Research Article | Open Access
W.-D. Richter, "Circle Numbers for Star Discs", International Scholarly Research Notices, vol. 2011, Article ID 479262, 16 pages, 2011. https://doi.org/10.5402/2011/479262
Circle Numbers for Star Discs
The notion of a generalized circle number which has recently been discussed for -circles and ellipses will be extended here for star bodies and a class of unbounded star discs.
Generalized circle numbers have been discussed for -circles in Richter [1, 2] and for ellipses in Richter . All these numbers correspond on the one hand to an area content property of the considered discs which is based upon the usual, that is, Euclidean, area content measure and a suitably adopted radius variable. On the other hand, they reflect a circumference property of the considered generalized circles with respect to a non-Euclidean length measure which is generated by a suitably chosen non-Euclidean disc. Several basic and specific properties of the circumference measure have been discussed in Richter [1–3]. We refer here to only two of them which are closely connected with each other by the main theorem of calculus. The first one is that the generalized circumference of the generalized circle coincides with the derivative of the area content function with respect to the adopted radius variable. The second one is that, vice versa, the area content of the circle disc equals the integral of the generalized circumferences of the circles within the disc with respect to the adopted radius variable. Integrating this way may be considered as a generalization of Cavalieri and Torricelli's method of indivisibles, where the indivisibles are now the generalized circles within the given disc and measuring them is based upon a non-Euclidean geometry. The far-reaching usefulness of this generalized method of indivisibles has been demonstrated in the work of Richter [1–4], where several applications are dealt with and where it was also shown that integrating usual, that is, Euclidean, lengths of the same indivisibles does not yield the area content, in general. In the present paper, we will prove that this method still applies when generalized circle numbers are derived for general star discs. In this sense, this paper deals with bounded and unbounded star discs. Notice that because we will not assume symmetry of the unit disc, distances will depend on directions in general.
To become more specific, let be a star body in , and let its area content be defined as usual by its Lebesgue measure. Furthermore, let us call the boundary of times the star disc the -circle of -radius , and denote it by . If we define the perimeter of by using different length measures, then we can observe different perimeter-to-(two-times--radius) ratios, and these ratios differ from the corresponding (area-content)-to-(squared--radius) ratio in general. If we choose, however, the length measure in a certain specific way, then the first ratio coincides with the second one for all . In the most famous case when is the Euclidean disc and measuring circumference is based upon Euclidean arc-length, the common constant value of the two ratios is the well-known circle number .
If is the symmetric and convex -circle, centered at the origin and thus defining a norm then, according to Richter , the suitable arc-length measure is based upon the dual norm, that is, the norm which is generated by the -circle with satisfying the equation .
Similarly, if is an -circle, , corresponding to an antinorm (see Moszyńska and Richter ) then, according to Richter , the suitable arc-length measure is based upon the star disc with satisfying . The star disc corresponds to a specific semi-antinorm with respect to the canonical fan (see Moszyńska and Richter ).
The situation for ellipses has been discussed in Richter . If is an elliptically contoured disc and its boundary then the suitable arc-length measure on the Borel -field of subsets of the ellipse is based upon the disc . Note that again and correspond to dual norms.
The arc-length measure used for defining the -generalized circle number allows both for and for the same additional interpretation in terms of the derivative of the area content function with respect to the -radius variable. This way, the notion of the -generalized circumference of the -circle was introduced first in Richter  under more general circumstances and motivated there by several of its applications. Several geometric interpretations of this notion in cases of special norms and antinorms have been discussed so far. As just to refer to a few of them, let us recall that this notion is a basic one for the generalized method of indivisibles, that it allows to prove the so-called thin layers property of the Lebesgue measure and to think of a certain mixed area content in a new way and that it is closely connected with the solution to a certain isoperimetric problem. From a technical point of view, a basic difference between the two situations is that in the convex case, one uses triangle inequality for showing convergence of a sequence of suitably defined integral sums, and that one makes use of the reverse triangle inequality from Moszyńska and Richter  for proving such convergence if the -generalized circle corresponds to an antinorm.
The arc-length measure used for defining ellipses numbers has also an interpretation in terms of the derivative of the area content function but with respect to a generalized radius variable corresponding to . The common notion behind the different definitions of a generalized radius variable discussed so far in the literature is that of the Minkowski functional (or gauge function) of a star body, but looking onto the motivating applications, for example, from probability theory and mathematical statistics, let it become clear that further generalizations are desirable in future work.
Here, we start our consideration with the definition of the -generalized circumference of an -circle corresponding to a star body and will discuss in Section 2 both its general geometric meaning and its specific interpretation either when is generated by an arbitrary norm or by an antinorm of special type.
It should be mentioned here that the Minkowski functional of a star body generates a distance which is not symmetric in general, that is, it does not assign a length in the usual sense to a generalized circle but a certain directed length.
Definition 1.1. Let be the Lebesgue measure in a star body, and the corresponding area content function. Then, will be called the -generalized circumference of or the -generalized arc-length of the boundary of , .
It follows from the properties of the Lebesgue measure that The representation
may be understood as a generalized method of indivisibles for the Lebesgue measure, where the indivisibles are multiples of the boundary of and measuring their circumferences is based upon . Equations (1.2) may suggest on the one hand to call the -generalized circle number. On the other hand, one may consider at this stage of consideration a method of introducing generalized circle numbers which follows basically the idea of the main theorem of calculus being rather elementary if not even trivial. However, the papers of Richter [1–4] which are closely connected with this approach allow a new look onto a class of geometric measure representations or, similarly, onto a class of stochastic representations which are quite fruitful for many applications. Several of these applications, especially in probability theory and mathematical statistics, are discussed therein.
Clearly, there is always a necessity to give a geometric or otherwise mathematical interpretation of the circumference . In other words, one naturally looks for a geometry such that the arc-length of with respect to this geometry coincides with the -generalized circumference . The non-Euclidean geometries being identified in this way may be considered as geometries “being close to the Euclidean one” as those were discussed in Hilbert  in connection with his fourth problem. If we can uniquely identify a geometry such that the arc-length measure of , , which is based upon the geometry's unit ball , satisfies then we can observe already the nontrivial situation that At such stage of investigation, it will then be already much more motivated that the area content of the unit star, , is called the -generalized circle number, .
2. Star Bodies
A subset from is called a star body if it is star-shaped with respect to the origin and compact and has the origin in its interior. A set of this type has the property that for every there exists a uniquely determined such that , where denotes the boundary of the set . This equals the value of the Minkowski functional with respect to the reference set
at any point . The function is often called the gauge function of (see, e.g., in Webster ) and coincides, for , with the reciprocal of the radial function (see, e.g., in Thompson  and Moszyńska ),
With a star body , the pair may be considered as a generalized Minkowski plane. The star disc and the star circle of -radius will be defined then by and , respectively. The set will be called the unit star in this plane.
Let be another star disc in which will be specified later. Whenever possible, we may define the -arc-length of the curve as follows.
Definition 2.1. If denotes a successive and positive (anticlockwise) oriented partition of , then the positive directed -arc-length of is defined by if the limit exists for and is independent of all described partitions of with tending to zero as .
Using triangle inequality or its reverse, one can show that if is a norm or antinorm then taking the limit may be changed with taking the supremum or the infimum, respectively. Notice that because is in general not a symmetric function, the orientation in the partition may have essential influence onto the value of and is, therefore, assumed here always to be positive.
For studying , let a parameter representation of the unit--circle be given by . Later on, we will assume that is a.e. differentiable. From the relation
it follows that
In other words, with the notation , we have
This motivates the following definition which generalizes more particular notions from earlier considerations.
Definition 2.2. For an arbitrary star body , the -generalized sine and cosine functions are respectively.
Notice that there is an elementary geometric interpretation of these generalized trigonometric functions when one considers a right-angled triangl with and as follows. The -generalized sine and cosine of the angle between the directions of the positive -axes and the line through the points and are
respectively. These functions satisfy
generalizing the well-known formula .
Definition 2.3. The -generalized polar coordinate transformation is defined by
Let us denote the quadrants in in the usual anticlockwise ordering by up to .
Theorem 2.4. The map is almost one-to-one, for , its inverse is given by and its Jacobian satisfies
Proof. The proof follows that of Theorem 8 in Richter  and makes essentially use of the fact that the derivatives of the -generalized trigonometric functions and allow the representations
Using -generalized polar coordinates, we can write
We assume from now on that is positively homogeneous, put
for sufficiently thin partition and . We get in the limit, which was assumed in Definition 2.1 to be uniquely determined
It follows from the proof of Theorem 2.4 that
that is, with
The following lemmas and corollaries represent certain steps towards a reformulation of formula (2.20).
Lemma 2.5. In the case of their existence, the partial derivatives and of the function satisfy the representation
Proof. It follows from the relation that the partial derivatives and satisfy the equation system Solving this differential equation system, we get Hence,
Let be a -matrix and . Clearly, multiplying the set by the matrix causes an anticlockwise rotation of through the angle . Hence, if is a star disc, then is a star disc too.
Corollary 2.6. For positively homogeneous , differentiable , formula (2.20) may be rewritten as
Remark 2.7. The plug-in version of the gradient coincides with the image of the gradient after changing Cartesian with -generalized polar coordinates, .
Proof. Changing Cartesian coordinates with -generalized polar coordinates , we have and . Starting from and the analogous one for , and taking into account that it follows
Remark 2.8. For positively homogeneous and differentiable Minkowski functional of the star disc , formula (2.20) may be rewritten as
Definition 2.9. A star body and a star disc satisfy the rotated gradient condition if
Let us notice that at the point from , the gradient is normal to the level set of . The following lemma is a consequence of the above consideration.
Lemma 2.10. For a star body and a star disc satisfying the rotated gradient condition (2.34), the positive directed -arc-length of allows the representation
We consider now the area function
where denotes the area content of the unit disc . The derivative of the area function satisfies obviously
The following theorem has thus been proved.
Theorem 2.11. If the star body and the star disc satisfy the rotated gradient condition (2.34), then that is, the -generalized circumference of coincides with the positive directed -circumference of .
If relation (2.38) holds, then Consequently, the ratios and satisfy the relations In this sense, the geometry and the arc-length measure generated by fulfill our expectations. The following definition is thus well motivated if a star body and a star disc are chosen in such a way that the limit in Definition 2.1 is uniquely determined, is positively homogeneous, is a.e. differentiable, and the rotated gradient condition (2.34) is satisfied.
Definition 2.12. (a) The properties of the star bodies , which are expressed by the equations (a) and (c) in (2.40) are called the area content and the -generalized circumference properties of the discs, respectively.
(b) The quantity is called the -generalized circle number of the star bodies .
We may write now (2.40) as Notice that the circle number function assigns a generalized circle number to any star body satisfying assumption (2.34). The restrictions of this function to -balls or axes aligned ellipses were considered in Richter [1–3].
Example 2.13. Here, we consider a first, rather general case, where the rotated gradient condition (2.34) is satisfied. Let and denote a (primary) -norm in and the corresponding dual one, respectively. It is proved in Yang  that
Hence, if is a convex body, that is, is a (primary) norm, and if
is the unit ball with respect to the corresponding dual norm, then the condition (2.34) is satisfied.
For determining the actual value of a generalized circle number we may refer, for example, to Pisier , where volumes of convex bodies are dealt with. Alternatively, one may use -generalized polar coordinates for making the respective calculations in given cases. The particular results for -circles with and of axes aligned ellipses as well as the corresponding generalized circle numbers have been dealt with in Richter [1, 3].
Example 2.14. We consider now the nonconvex -circles with . Such generalized circles correspond to antinorms. A suitable arc-length measure for measuring is based upon the star disc with satisfying . The star discs are closely related to specific semiantinorms with respect to the canonical fan. The corresponding generalized circle numbers have been determined in Richter . As because this was done without referring explicitely to (2.34), we may state here the following problem.
Problem 1. Give a general description of sets satisfying condition (2.34) for sets being generated by antinorms.
As was indicated in Richter , -generalized circle numbers for may occur, for example, within certain combinatorial formulae. Notice further that the reciprocal values of the coefficients of the binomial series expansion
are just the generalized circle numbers corresponding to the nonconvex -circles. One could also ask for a (possibly elementary geometric?) explanation of this fact.
3. Unbounded Star Discs
In this section, we consider a class of (truncated) unbounded Orlicz discs. More generally than in the preceding section, a star-shaped subset of is called a star disc if all its intersections with balls centered at the origin are star bodies. The boundary of a star disc is called a star circle. Notice that a star circle is not necessarily bounded. Special sets of this type will be studied in this section. To be more specific, let us consider, for arbitrary , the function
which denotes a semi-antinorm, and the -generalized circle
The pairs of straight lines and represent asymptotes for the circle as or , respectively. The intersection point of the -circle with the line is for positive coordinates .
Let further denote the -generalized circle of -generalized radius . It is the boundary of the unbounded -generalized disc of -generalized radius ,
one may call a two-dimensional Orlicz antiball corresponding to the Young function . The disc is a star-shaped but noncompact set and, therefore, not a star body. For any one may think of as the value of the Minkowski functional with respect to the reference set . The area content and the Euclidean circumference of the unit -circle are obviously unbounded. That is why we consider from now on truncated -circles. To this end, let us introduce truncation cones
where denotes Euclidean norm, , is chosen according to and .
The question of interest is now whether we may define in a reasonable way circle numbers for the truncated -discs , the boundaries of which are the -circles of -radius , where
To this end, let be an arbitrary successive anticlockwise-oriented partition of the truncated circle satisfying and . We consider the sum
and observe that due to the reverse triangle inequality it decreases monotonously as
According to the symmetry of , the following remark is justified.
Remark 3.1. For , the -arc-length of the truncated circle is defined as
If denotes an arbitrary parameter representation of the truncated circle , then
Let us denote the usual Euclidean area content of the truncated circle disc by .
Lemma 3.2. Let for arbitrary the number be uniquely defined by the equation . Then,
Proof. With it follows from the above formulae that Changing variables causes a change of the limits of integration: Assuming now , or equivalently , it follows that Hence, Now, what about the area content of the truncated circle ? The -generalized standard triangle coordinate transformation from Richter  is defined by Because of it follows That is, Changing Cartesian with standard triangle coordinates in the integral we get Substituting , it follows that Hence, the lemma is proved.
Remark 3.3. Because of ,
The following corollary and definition are now quite obvious and well motivated.
Corollary 3.4. For arbitrary , the truncated star discs have the area content and -generalized circumference properties and , respectively, from which it follows immediately
Remark 3.5. One may think of (3.26) as reflecting a generalized method of indivisibles for each , where the truncated circles are the indivisibles and measuring them is based upon the geometry generated by the disc .
Definition 3.6. For arbitrary , the quantity will be called the circle number of the truncated circle .
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Copyright © 2011 W.-D. Richter. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.