Abstract
Two-dimensional steepest descent curves (SDC) for a quasiconvex family are considered; the problem of their extensions (with constraints) outside of a convex body is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on . These regions are bounded by arcs of involutes of and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self-contracting sets (with opposite orientation) are considered: necessary and/or sufficient conditions for them to be subsets of SDC are proved.
1. Introduction
Let be a smooth function defined in a convex body . Let in . A classical steepest descent curve of is a rectifiable curve solution to Classical steepest descent curves are the integral curves of a unit field normal to the sublevel sets of the given smooth function . We are interested in “generalized” steepest descent curves that are integral curves to a unit field normal to a nested family of convex sets (see Definition 5); will be called a quasiconvex family as in [1]. Sharp bounds about the length of the steepest descent curves for a quasi convex family have been proved in [2–4]. The geometry of these curves, equivalent definitions, related questions and generalizations have been studied in [5–9].
In the present work generalized steepest descent curves for a quasiconvex family (SDC for short) are defined as bounded oriented rectifiable curves , with a locally Lipschitz continuous parameterization , with ascent parameter, satisfying is the scalar product in . Let ordering be chosen on , according to the orientation; let us denoteIn [8, Theorem 4.10], the SDC are characterized in an equivalent way as self-distancing curves, namely, oriented () continuous curves with the property that the distance of to an arbitrarily fixed previous point is not decreasing:Thus steepest descent curves are self-distancing curves and both denoted SDC. In [8] self-distancing curves are called self-expanding curves. With the opposite orientation these curves have been also introduced, studied, and called self-approaching curves (see [2]) or self-contracting curves (see [7]).
In our work we are interested in the behaviour and properties of a plane SDC beyond its final point . One of the principal goals of the paper is to show that conditions (2) and (4) imply constraints for possible extensions of curve beyond ; these constraints are written as bounding regions for the possible extensions of .
An important property that will be used later is the property of distancing from a set .
Definition 1. Given a set , an absolutely continuous curve , has the distancing from property if it satisfies
Let us outline the content of our work. In Section 2 introductory definitions are given and covering maps for the boundary of a plane convex set, needed for later use, are introduced. In Section 3 the involutes of the boundary of a plane convex body are introduced and some of their properties are proved.
In Section 4 plane regions depending on the convex hull of have been defined; these regions fence in or fence out the possible extensions of . The boundary of these sets consists of arcs of involutes of convex bodies, constructed in Section 3. As an application, in Section 4.1 the following problem has been studied: given a convex set , , , is it possible to construct joining to , satisfying the distancing from K property? Minimal properties of this construction have been introduced and studied. In Section 5 sets of points more general than SDC are studied. A set (not necessarily a curve) of ordered points satisfying (4) will be called self-distancing set; see also Definition 2; with the opposite order, was called in [6] self-contracting set and many properties of these sets, as only subsets of self-contracting curves, were there obtained. Another goal of the paper is the solution to the following question: given a self-distancing set does a steepest descent curve exist? In Section 5 examples, necessary and/or sufficient conditions are given when consists of a finite or countable number of points and/or steepest descent curves .
In the present work the two-dimensional case is studied. Similar results for the -dimensional case are an open problem stated at the end of the work.
2. Preliminaries and Definitions
Let A nonempty, compact convex set of will be called a convex body. From now on, will always be a convex body not reduced to a point. and denote the interior of and the boundary of , denotes its length, is the closure of , will be the smallest affine space containing , and and are the corresponding subsets in the topology of . For every set , is the convex hull of .
Let ; the normal cone at to is the closed convex cone:When , then reduces to zero.
The tangent cone or support cone of K at a point is given by
In two dimensions cones will be called sectors.
Let be a convex body and let be a point. A simple cap body isCap bodies properties can be found in [10, 11].
2.1. Self-Distancing Sets and Steepest Descent Curves
Let us recall the following definitions.
Definition 2. Let us call self-distancing set a bounded subset of , linearly ordered (by ), with the property
The self-distancing sets have been introduced in [6] with the opposite order. If a self-distancing set is a closed connected set, not reduced to a point, then it can be proved that is the support of a steepest descent curve (see [8, Theorem 4.10, Theorem 4.8]) and it will also be called a self-distancing curve .
The short name SDC will be used both for self-distancing curves and for steepest descent curves in all the paper.
Definition 3. Let be a convex body; will be called a self-distancing curve from (denoted ) if (i)is a self-distancing curve,(ii),(iii) has the propertyWhen (ii) does not hold, that is , will be called a deleted self-distancing curve from .
Remark 4. Let be , since has an absolutely continuous parameterization ([8, Theorem 4.10, Theorem 4.8]), thus property (11) for is equivalent to (5).
If is SDC and then is . That is, the tangent vector to is in the normal cone at to the related convex set . This condition is a generalization of the classicals steepest descent curves that are integral curves to a unit field normal to smooth quasiconvex families.
Nested families of convex sets have been introduced and studied by de Finetti [12] and Fenchel [1]. Let us recall some definitions.
Definition 5. Let be a real interval. A convex stratification (see [12]) is a nonempty family of convex bodies , , linearly strictly ordered by inclusion (, ), with a maximum set () and a minimum set ().
Let be a convex stratification. If for every the property holds, then as in [1], will be called a quasiconvex family.
An important quasiconvex family associated with a continuous self-distancing curve from , : is , where The couple is special case of Expanding Couple, a class introduced in [8].
Remark 6. If , then for all the curve is a self-distancing curve from the convex hull of the set .
This fact is a direct consequence of the following.
Proposition 7 (see [8, Lemma 4.9]). Let , . Ifthen the same holds for every .
The statement of the previous proposition holds if large inequalities are replaced by strict inequalities everywhere.
2.2. The Support Function of a Plane Convex Body
Let be a convex body not reduced to a point.
For a convex body , the support function is defined aswhere denotes the scalar product in . For , , let and ; it will be denoted if no ambiguity arises.
For every there exists at least one point such thatthis means that the line through orthogonal to supports . For every let be the set of such that (16) holds. Let be the set of all satisfying (16). If is strictly convex at the direction then reduces to one point and it will be denoted by .
Definition 8. The set valued map , is the generalized Gauss map; is a vertex on iff is a sector with interior points. The set valued map is the reverse generalized Gauss map; is a closed segment, possibly reduced to a single point, and it will be called 1-face when it has interior points.
Let be the covering map
Let , and let , (, ) be the parametric representations of depending on the arc length counterclockwise (clockwise) with an initial point (not necessarily the same). Let us extend and by defining similarly for .
Let us fix , , , and .
For later use, we need to have ; this can be realized by choosing suitable initial points for the parameterizations and .
Then
The maps are covering maps.
The initial parameters will be ( are the back images of , resp.). Let . Let us define, for :if Similarly, let us define for :if The function is increasing in and right continuous and with left limits (so called cadlag function). Similar properties hold for . Let us recall that a cadlag increasing function , , has a right continuous inverse defined as Let be the right continuous inverse of . Let be the opposite of the right continuous inverse of .
Let us introduce for simplicity
Let be the support function of .
It is well known ([13]) that if is (i.e., , with positive curvature), then is and the counterclockwise element arc of is given by is the positive radius of curvature; moreover the reverse Gauss map is a 1-1 map given by
The previous formula also holds for an arbitrary convex body, for every such that is reduced to a point; see [10]. Let us recall that a real valued function is called semiconvex on when there exists a positive constant such that is convex on . From (29) the function is convex on ; thus is semiconvex. In the case that is an arbitrary convex body, by approximation arguments with convex bodies (see [11]) it follows that the support function of is also semiconvex. As consequence is Lipschitz continuous, it has left (right) derivative (resp. ) at each point, which is left (right) continuous. Moreover at each point the right limit of is and the left limit of is ; see [14, pp. 228].
It is not difficult to show (from (30), with a right limit argument) that, for an arbitrary convex body, for , the formulaholds. Similarly the formulaholds.
If is not strictly convex at the direction then is not differentiable at andIf let us define as the set of points of between and according to the counterclockwise orientation of and as the set of points between and , according to the clockwise orientation; denote their length.
Remark 9. It is well known that a sequence of convex body converges uniformly to if and only if the corresponding sequence of support functions converges in the uniform norm; see [11, pp. 66]. Moreover as the two sequences of the end points of a closed counterclockwise oriented arc of converge, then the sequence of the corresponding arcs converges to a connected arc of and the sequence of the corresponding lengths converges too.
Proposition 10. Let be a convex body and its support function; then
Proof. For every convex body not reduced to a point the function is defined everywhere and satisfies the weak form of (29); namely,Using the fact that is Lipschitz continuous, integrating by parts (36), the formulaholds. Thus with constant. Passing to the right limit, the equality holds. Formula (34) follows, by computing , using the previous equality. Similarly (35) is proved.
3. Involutes of a Closed Convex Curve
Definition 11. Let be an interval. A plane curve is convex if at every point it has right tangent vector and arg is not decreasing function.
Let be the arc length parameterization of a smooth curve; the classical definition of involute starting at a point of the curve is
Let us notice that is the arc length of the curve, not of the involute; if , then the starting point of the involute coincides with the starting point of the curve. It is easy to construct an involute of a convex polygonal line (even if classical definition (40) does not work) by using arcs of circle centered at its corner points; moreover the involute depends on the orientation of the curve.
In this section, involutes for the boundary of an arbitrary plane convex body , not reduced to a point, will be defined. The assumption that is an arbitrary convex body is needed to work with the involutes of the convex sets, not smooth, studied in Section 4.
Let ; let be a fixed point of ; can be the clockwise parameterization of or the counterclockwise parameterization. Since there exist two orientations, then two different involutes have to be considered. As noted previously one can assume that the parameterizations of have been chosen so that .
Definition 12. Let one denote by the left involute of starting at corresponding to the counterclockwise parameterization of and by the right involute corresponding to the clockwise parameterization. When one needs to emphasize the dependence on of involutes, they will be written as , .
Remark 13. Let us notice that if is a plane reflection with respect to a fixed axis then This relation allows us to prove our results for the left involutes only and to state without proof the analogous results for the right involutes.
Theorem 14. Let one fix the initial parameters , , , and . The left and the right involutes of a plane convex curve starting at , boundary of a plane convex body with support function , are parameterized by the value related to the outer normal to , as follows:
Proof. In the present case there is a 1-1 mapping between and ; from (29), it follows that then, changing the variable with in (40), with elementary computation, (42) is obtained (since and (30) holds). Formula (43) follows from (32) and (35).
For an arbitrary convex body in place of (30), formulas (31) and (32) have to be used.
Definition 15. Let be a plane convex body; let The left involute of starting at will be defined assimilarly if , , the right involute starting at will be defined as
From (46) and (34) it follows thatsimilarly from (47), (35) it follows that
Let us notice that in (48) and (49) the same parameter is used, but with different range; it turns out that is counterclockwise oriented; instead is clockwise oriented; .
Remark 16. The following facts can be derived from the above equations: (i)since is Lipschitz continuous for every convex body , then the involute is a rectifiable curve;(ii) and(iii)if is a vertex of then , for , lies on an arc of circle centered at with radius ;(iv)the involute (42) satisfies
Lemma 17. Parameterization (48) of the involute is 1-1 in the interval ; moreover, except for at most a finite or countable set of values , (corresponding to the 1-face of ), is differentiable andfurthermore has left and right derivative with common direction at .
Proof. By differentiating (48) and using (34), equality (52) is proved. Similar argument, at , proves that is the common direction of the left and right derivatives.
Remark 18. Let , , , be left involutes of . Since then they will be called parallel curves. Moreover, by (51), and will also be called parallel.
Theorem 19. If is the arc element of the involute then is continuous and invertible in with continuous inverse . Moreoverthe involute is a convex curve with positive curvature a.e.: is everywhere, andMoreover the following properties hold. (i)For every the right derivative exists everywhere and it is a decreasing cadlag function.(ii) has everywhere right derivative given by
Theorem 20. Let be a sequence of plane convex bodies which converges uniformly to , , ; then the corresponding sequences of left involutes converge uniformly to in compact subsets of ; moreover the corresponding sequence of their derivatives (with respect to the arc length) converges uniformly to .
Proof. By Remark 9 the sequence of functions converges to . From (54) the arclengths of the left involutes converges uniformly in compact subsets of to the arc length of ; from (56) the same fact holds for their derivatives.
Let us consider the arc of the involute and the set valued map (Definition 8). Let the union of segments joining the points of with the corresponding points on .
Definition 21. If the tangent sector to has an opening less than or equal to as in Figure 1, then is convex; let one define If is not convex then let us consider . Let us notice that is an open segment with end points , with , . Let us define , with such that (see Figure 2) is orthogonal to , . Let be the smallest satisfying . Clearly .
For the right involutes a value is defined similarly, with , such that the line orthogonal to supporting at is tangent to the right involute at (see Figure 3) where is the point , written as for short.
Theorem 22. Let be the left involute starting at on the boundary of a plane convex body ; then (i)the left involute has the distancing from property for but is not for ;(ii)the curve is ;(iii)for the distance function is strictly increasing for ;(iv)if , then is not decreasing for and for .
Proof. As is rectifiable, then the function is an absolutely continuous function for , and from (52) for the last inequality holds since is the outer normal to at . Moreover the previous inequality is strict for all if , and it is also a strict inequality for and . This proves (iii) and (iv). Then (i) follows from (iii) and Definition 1 of distancing from K property for a curve. To prove (ii) let us recall that SDC satisfies (2); then one has to prove that the angle at between the vector , , and , the tangent vector at , is greater than or equal to ; this is equivalent to show that the half line through and orthogonal to supports at the arc of from to . By Definition 21 this is the case for all between and .
Corollary 23. The left involute of the boundary of a plane convex body is a self-distancing curve from for ; similarly right involute (49) is a self-contracting curve from for .
Proof. From (i) of Theorem 22 the left involute is a curve such that the distance of its points from all is not decreasing; (ii) of the same theorem proves that it is a . Let us recall that a self-contracting curve is a self-distancing curve with opposite orientation.
Theorem 24. Let be a plane convex body not reduced to a single point and let be the initial parameters. Let be an of the left involute starting at , and let be an arc of the right involute ending at ; then there exists only one point which belongs to both arcs andwith
Proof. For simplicity, first let us prove the existence of assuming that . With the assumed conditions, defined by (30) is a parameterization of .
Let and let be the first common point of the half line and of . Moreover, let be the function satisfyingLet First the following sentence will be proved.
Claim 1. belongs to iff the equalityholds for some , .
Proof of Claim 1. If (68) holds, then Thus Thus is on both arcs of involutes and the other way around.
Our aim is to prove that there exists such that (68) holds. For this goal we prove next Claims 2 and 3.
Claim 2. The following facts hold in : (i) is continuously differentiable and ;(ii). Proof of Claim 2. Let us prove that and satisfyLet us consider the triangle with vertices . As the angle between and is acute and the angle between and is obtuse. Thus (71) follows. By definition, solves (66); thus is the implicit solution toAs is negative by (71), then by Dini’s Theorem equation (73) has a solution satisfying As and (71) holds, then , and is strictly increasing and continuously differentiable.
Let us prove (ii).
The formulaholds. Let us notice that is parallel to ; thus by (52) On the other hand As the angle between and is acute, then last term in the above equalities is positive; thus the derivative in the left hand side of (76) is positive and (ii) of Claim 2 follows.
Claim 3. In the interval the function has values smaller than and greater than .
Proof of Claim 3. The angles and have been introduced in Definition 21. For simplicity will be denoted with . Let us consider the convex set bounded by and by the polygonal line with vertices ; see Figure 3.
Clearly the inequalities hold. As using the previous inequalities, one obtains Let us show now thatholds.
Let be the half line with origin and direction ; crosses the arc in a first point , with . Then The half line meets the arc in a point and
Property (iii) of Theorem 22 implies that the arc of the left involute after lies outside of the circle centered in and with radius . Similar property for the right involute implies that the arc of the right involute joining to lies in the circle with center and radius ; thus the straight line tangent to at meets the arc in and in . Therefore Inequality (82) is proved.
The intermediate values theorem implies that there exists such that (68) holds. Claim 1 implies that so the right involute and the left involute meet each other in one point and (64) is proved with , .
By approximation argument the same result holds for an arbitrary convex body .
Let us prove now that the point is unique. Let us argue by contradiction. Let be two distinct points on , with on and ; then since is a distancing curve from , and since is a contracting curve to , therefore all the points on the arc of and of between and have the same distance from ; thus, between and , and (arc of involutes of a same convex body ) coincide with the same arc of circle centered at ; this implies that reduce to the point , which is not possible for the assumption.
Definition 25. Let . Let on the contact set on the “left” (right) support line to through . If the contact set is a 1-face on these support lines, then and are identified as the closest ones to . The triangle is counterclockwise oriented.
Theorem 26. For every let one consider the left involutes and the right involutes parameterized by their arc length . The maps are 1-1 maps.
Proof. Assume, in the proof, that are fixed. Let . The tangent sector to the cap body with vertex z has two maximal segments and on the sides that do not meet (except at the end points and ). Let be such that , and let be such that Let ; let . From (50) and from the definition of left involute (46) (with in place of , in place of , and in place ) holds; thus the map is surjective. Moreover the map is also injective, since the left involutes do not cross each other since they are parallel (see Remark 18). Similar proof holds for the right involutes.
Let be the starting point of the left involute through , defined in the previous theorem; similarly let be the starting point of the right involute through . Let us notice that and meet each other in a countable ordered set of points.
3.1. -Fence and -Fence
Definition 27. Let be a convex body in , , , , , . Let , and let . Let be the first point where the two involutes cross each other (see Theorem 24). Let one define will be called the -fence of at .
Let us notice that and are two convex bodies in common with the segment only.
From Theorem 26 the starting point () of a left (right) involute is uniquely determined from any point of the involute. The arc of the points on the left (right) involute between the starting point and will be denoted by () or () for short. For let us denote with the oriented arc of the left (right) involute between and .
Let us introduce now other regions which are bounded by left and right involutes.
Let us fix the initial parameters .
Definition 28. Given , let () be the left (right) involute through with starting point () and let be as in Definition 25. Let satisfying . Let be the smallest angle for which . Let one consider the parameterization (46); let one defineIf does not cross the open segment , the region is an open set bounded by the convex arc of left involute , the segment , and the convex arc of : ; otherwise let be the nearest point to where crosses the open segment ; the region is an open set bounded by the arc , the segment , and . Similarly let us define .
are open and bounded sets. Let us define is an open, bounded, connected set. will be called the -fence of at .
Remark 29. If is the first crossing point of and and , then .
Let us conclude this section with the following result, which follows from Theorem 20.
Theorem 30. Let be limit of a sequence of convex bodies , , and . Then Moreover if , , then
4. Bounding Regions for SDC in the Plane
Let us assume that is the end point of one of the following sets:(a)a steepest descent curve ;(b): a self-distancing curve from a convex body ; see Definition 3. The following questions arise: can one extend , beyond ? Which regions delimit that extension? Which regions are allowed and which are forbidden?
Lemma 31. Let . If then the arc of the left involute to ending at is contained in . Similarly if , then .
Proof. Since , by (93) there exist , such that Then the arc is parallel to an arc of the left involute (through ) for . Then any left tangent segment to from a point of is contained in the left tangent segment from the corresponding point of .
Lemma 32. Let and let . There are two possible cases: (i)if the right involute ending at does not cross the tangent segment or it crosses at a point , then in both cases ;(ii)if the right involute ending at crosses the tangent segment at a point , then .
Proof. Since the starting point of the right involute ending at is on , the distance from to a point of the left involute is not decreasing; see (iv) of Theorem 22; similarly the distance from to a point of is not decreasing. In case (i) the arc has its end points in and by the above distance property it can not cross two times the left involute; then it can not cross the boundary of ; therefore ; similarly in case (ii) the arc can not cross the boundary of at most in ; therefore all the points of this arc, except to that , belong to .
From the previous lemma the following follows.
Theorem 33. Let . The following inclusions hold: (a)if , then(b)if , then(c)if , then
Proof. By Lemma 31 the left involute that bounds is inside ; then (98) is proved. Inclusion (99) is proved similarly. Let and let us consider ; then in case (i) of Lemma 32 also the open arc of the right involute is inside . Besides ; then (100) is trivial. In case (ii) of Lemma 32 the open arc is inside . On the other hand is inside and by (99) the arc is in . Similar arguments hold if . Then (100) holds in this case too.
Lemma 34. Let . Let be polygonal deleted with end point . Then
Proof. To prove (101), let us assume, by contradiction, that has a point . With no loss of generality it can be assumed that and Then, is the end point of a segment , where and on . As , then there exists an involute through which is a piece of the boundary of (to fix the ideas it is assumed that it is the left involute ). Let us consider so that the tangent vector to at satisfies As is inside the orthogonal angle centered in with sides and , then Then as for sufficiently small, and at the curve has tangent vector that satisfies contradicting the fact that has the distancing from property (5). This proves (101).
If , with , also the inclusions hold. Then (102) is obtained by approximation Theorem 30.
Theorem 35. Let be a convex body and let be , . Then
Proof. Let us choose a sequence . Let us fix the arc . By [8, Theorem 6.16], is limit of polygonal with end point . From Lemma 34, these polygonal are enclosed in ; then holds too. Inclusion (108) is now obtained from the limit of the previous inclusions and by the approximation Theorem 30.
Theorem 36. Let be a convex body not reduced to a point. If is a self-distancing curve from with starting point , then
Proof. Let be the first crossing point of the left and right involutes of starting at . Then By contradiction, if has a point , then, by Theorem 35, the following inclusion holds: since, by the distancing from property, has in common with only the starting point then the inclusion holds too. Moreover by (100) the set has positive distance from the ; then has a positive distance from . This is in contradiction with .
Corollary 37. Let be and let ; then
Proof. Since is a self-distancing curve from and (see [8, (i) of Lemma 4.6]), then Theorem 36 applies to with .
Definition 38. Let be . If , with let
For , let be the cap body, introduced in (9). Next theorem shows the principal result on bounding regions for arcs of SDC .
Theorem 39. Let be a convex body and let be . If , with then
Proof. First let us notice that has the distancing from and from the set point property; thus by Proposition 7 it has the distancing from property. Then inclusion (116) follows from Theorems 35 and 36.
Let us conclude the section with the following inclusion result for -fences.
Theorem 40. Let be two convex bodies not reduced to a point, . Let . Then
Proof. The boundary of consists of two arcs of the left and right involutes of starting at . By Corollary 23 they are , and then they are ; therefore by Theorem 36 they cannot intersect the boundary of .
4.1. Minimally Connecting Plane Steepest Descent Curves
Given a point , the segment joining it with its projection on is which minimally connects the two points.
This subsection is devoted to consider when it would be possible to connect a given point on the boundary of a plane convex body , with an arbitrarily given point , by using a steepest descent curve . Let us denote with the class of the curves starting at and ending at .
Definition 41. Let be SDC with end point and let be SDC with starting point ; let us denote by the curve joining with in the natural order, if it is curve.
Theorem 42. Let , . Then iffIf (118) holds, there exist at most two , such that the following properties are true:
Proof. Let . From (110) of Theorem 36, since , then (118) follows.
Let us prove now that (118) is sufficient. Let us notice that can be divided into four regions , , , and (see Figure 4) defined as follows: (i)the closed normal sector is the angle bounded by the two half lines tangent at to the left and right involute , , respectively; this angle can be reduced to an half line, starting at ;(ii)let be the first crossing point between and ; see Theorem 24; is to , which will be a point following ; after the involute is no more ; see (i) of Theorem 22. Let us change after with , the left involute of at . Let us define as the first intersection point of with , and let It is not difficult to see that . Changing the left with the right, and the point can be constructed. Let be the union of the arc with the plane open region bounded by the segment , the arc , and the arc ; let be the union of the arc with the plane open region bounded by the segment , the arc , and the arc ;(iii)let be the remaining region; that is, .Let . On the oriented curve there are two points so that their tangent lines contain . Let be the first tangency point.
Let be the intersection points of the half line starting at and containing , with and with , respectively; see Figure 4.
Under assumption (118), belongs to one of the four regions ; let us prove now (119), (120) in the four corresponding cases. (1)If , then let be the segment . Then (119) and (120) are trivial.(2)Let . Let The curve is , since the normal lines at all the points on the segment have the same directions and support up to ; then is and joins with . Similarly is defined: Thus is nonempty and contains at least the two elements . Let us consider the connected closed curve Let , and let be a continuous parameterization of . Let us project from the curve on and let be this projection. That is, for , let and let Clearly is a closed connected subset of containing and the segment . Thus contains at least one of the two connected components of joining with . Therefore the inclusions or (or both) hold. Assume that (126) holds and let be defined as in (122). Since, by construction of , the set is contained in , then Similarly if (127) holds, then with defined by (123). Then (119) is proved. It is not difficult to see that the region bounded by contains the convex region bounded by . Thus the bound holds; similar procedure can be used for left case. This proves (120).(3)Let ; the same argument as in case (2) can be carried on up to the inclusions (126) and (127). As in step (2), when case (126) holds, the curve can be constructed and is . Let us show that if then (127) cannot occur, so the curve can not to be constructed. Let us argue by contradiction. If (127) occurs, then let be the first point where crosses the half line . The point exists, since, under the assumption (127), . Then (Theorem 36) does not belong to the open segment . Then, from (127) (see Definition 27) Moreover is ; see Remark 6. Then by (131) it is . Let us consider the convex body . Since then is a deleted . From Theorem 39, with in place of , in place of , it follows that Let be where the right tangent from to crosses the arc . Let us notice that is also an arc of the left involute of at . Moreover is an arc of the left involute of starting at ; then it is parallel to the left involute of at until it crosses the sector ; it turns out that From the two previous inclusions a contradiction comes out since Then (127) cannot occur.(4)The case is similar to the previous one.The proof is complete.
Definition 43. Under the assumptions of Theorem 42, let one define the set of (possibly coinciding) as they are constructed in the proof of Theorem 42, which satisfy (119) and (120). These curves will be called minimally connecting steepest descent curves for the class .
Definition 44. Let be an absolutely continuous curve and let be a point of , with tangent vector . Let is half plane bounded by the normal line to at and it is defined almost everywhere in . For the curve (consisting of the points of following ) let us define the region: If , then it is a convex set.
If is SDC ( is ), then condition (2) (resp., (5)) implies that
Theorem 45. Let . Let be with first point . Necessary and sufficient conditions for the existence of a curve , self-distancing curve from , starting at and satisfying , are as follows: (a);(b)there exists such that ; moreover if (a) and (b) are satisfied, then .
Proof. (a) is necessary by Theorem 36. (b) is necessary by Theorem 42 and by (138) since . Conversely if (a) and (b) hold, let us define ; then (by definition of ) is ; thus, is too and is (see Remark 6).
5. Self-Distancing Sets and Steepest Descent Curves
A self-distancing set will be called SDC-extendible if there exists a steepest descent curve such that .
This section is devoted to investigate the following question: Can a self-distancing set be extended to a steepest descent curve ?
Let us call the family of SDC which extends . The following example shows that can be empty.
Example 46. Let us consider in a coordinate system xy the points The set is a self-distancing set not SDC-extendible.
Proof. By contradiction let ; then any point on the arc satisfies the inequalities That is, and Since , the arc consists of two points only, which is impossible.
Next theorem gives a necessary condition (141) in order to extend a finite self-distancing set to SDC; this condition is based on the bounding sets introduced in Section 3.1.
Let us define as the subset of consisting of the point and of the previous ones on (consistent with (3)).
Theorem 47. Let be a self-expanding SDC-extendible set; then for all such that , the inclusionholds.
Proof. Let ; then and is SDC. Then and (see [8, (i) of Lemma 4.6]); from Theorem 40, moreover and from Corollary 37, The previous inclusions prove (141).
Remark 48. In Example 46 it has been proved, in a simple way, that is not SDC-extendible. Another way to prove this fact is to check that condition (141) does not hold for the point ; let us notice that consists of a circular arc centered at with radius ; then it is easy to see that is in the interior of and (141) is not satisfied.
Let us show in the following example that (141) is not sufficient for a self-distancing set to be SDC-extendible.
Example 49. Let us consider in a coordinate system the points For , the set is a self-distancing set satisfying condition (141) not SDC-extendible.
Proof. It easy to see that is a self-distancing set. Moreover the initial piece of the left involute of starting at consists of a circular arc centered at of ray and amplitude (). Then and (141) is verified with , . Trivially (141) is verified also at . Let us prove now that is empty. By contradiction let . Let us consider . Since , have the same distance from , arguing as in Example 46, is a circular arc centered at from to . Since the arc has the distancing property from the segment , it is necessarily the arc of amplitude and not the complementary arc. Let . Since and , then Thus by Theorem 40 Since the segment is tangent to at and it has length , less than , the length of the arc , then This is in contradiction with Corollary 37 at the point .
Let us introduce definitions and preliminary facts needed to obtain necessary and sufficient conditions for the extendibility of a self-distancing set structured as follows.
Definition 50. Let one denote with a self-distancing set with a finite (or countable) family of closed connected components , ordered as the points of ; that is, if , , . Let be the first point and let be the last point of ; if they are distinct (i.e. does not reduce to a point) as noticed in the introduction ([6, Theorem 3.3] and [8, Theorem 4.10]), is SDC and it will be denoted by .
Lemma 51. Let be a self-distancing set. A necessary condition for is that for all components , which are curves , the factholds.
Proof. Let . Then . Then (147) follows from (138).
Definition 52. Let be a self-distancing set. A subfamily is called essential for if the factshold.
If , let us define essential for .
Let us start to study a self-distancing set with two closed connected components.
Lemma 53. Let be a self-distancing set and let be the segment joining . There are five possibilities: (p1)Let ; then is essential for .(p2)Let , ( is ); then a necessary and sufficient condition for the extensibility of ismoreover is essential for . (p3)Let be a SDC, ; then a necessary and sufficient condition for the extensibility of is moreover (see Definition 43).(p4)Let , ; then a necessary and sufficient condition for the extensibility of is that there exists SDC such thatthe related essential family is (153).(p5)If in cases (p2), (p3), and (p4) the corresponding necessary and sufficient conditions do not hold, then
Proof. Case (p1) is trivial; in case (p2) the inclusion (151) follows from Lemma 51 with in (147). It is also trivial that it is sufficient. Case (p3) follows from Theorem 42 with . Case (p4) follows from Theorem 45 with in place of , , in place of and in place of .
An easy sufficient condition to check if is extendible is the following.
Theorem 54. Let be a self-distancing set. Let . If (147) andhold, then and , which linearly and orderly connects , with the segments , is and it has minimal length in .
Proof. Let us argue by induction on the self-distancing set . The case is contained in Lemma 53, since the assumptions (147) and (155) are enough to get the corresponding assumptions in cases (p1), (p2), (p3), and (p4). Moreover in the case (p4) the curve is such that is the SDC of minimal length extending .
Let be the curve of minimal length extending . Since the normal sector to at coincides with the sector in the proof of Theorem 42, with in place of , in place of , then assumption (155) implies that case (1) of the proof of Theorem 42 occurs. It follows that is of minimal length in . Then, is of minimal length in .
Remark 55. Since, as noticed in [4, II, Section 2], the angle has opening less than , thus the related normal sector in (155) has opening greater than or equal to ; then checking that satisfies (155) is easier than checking that is outside of the -fence as in (152).
Lemma 51 and Theorem 54 give only necessary and only sufficient conditions, respectively, for the extensibility of self-expanding sets. Let us give definitions in order to get necessary and sufficient conditions.
Definition 56. Let be a self-distancing set. Let be defined by induction as follows.
is the essential family related to , as given by Lemma 53; if , the related to is defined as follows: (i)if then ;(ii)if , let one consider for all the essential family (see Lemma 53) related to (see Definition 50). Let .
Let us notice that is ordered by inclusion and , if it is nonempty, consists of curves at most.
Theorem 57. Let be a self-expanding set and let be the sequence (finite or countable) of the essential families associated with . Then iff the essential family is nonempty.
Proof. If there exists then, for all , ; thus by Theorem 42. Conversely if at each step the essential family , then, by definition, there exists a sequence of SDC such that and such that (i.e., at each step, is an extension of a previous one ) and ; see Definition 56. Then is well defined; obviously .
Open Problem. In the present work only two-dimensional problems are studied. In three (or more) dimensions the construction of boundary regions to SDC and to is open. The boundary regions should probably be constructed by using the space involutes of the geodesics curves on .
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work has been partially supported by INdAM-GNAMPA (2014).