Abstract

The notion of a generalized circle number which has recently been discussed for 𝑙2,𝑝-circles and ellipses will be extended here for star bodies and a class of unbounded star discs.

1. Introduction

Generalized circle numbers have been discussed for 𝑙2,𝑝-circles in Richter [1, 2] and for ellipses in Richter [3]. 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 [13]. 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 [14], 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 𝑆 be a star body in IR2, and let its area content be defined as usual by its Lebesgue measure. Furthermore, let us call the boundary of 𝜚 times the star disc 𝑆 the 𝑆-circle of 𝑆-radius 𝜚, 𝜚>0 and denote it by 𝑆(𝜚). If we define the perimeter of 𝑆 by using different length measures, then we can observe different perimeter-to-(two-times-𝑆-radius) ratios, and these ratios differ from the corresponding (area-content)-to-(squared-𝑆-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 𝜚>0. In the most famous case when 𝑆 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 𝜋.

If 𝑆 is the symmetric and convex 𝑙2,𝑝-circle, centered at the origin and thus defining a norm then, according to Richter [1], the suitable arc-length measure is based upon the dual norm, that is, the norm which is generated by the 𝑙2,𝑝-circle {(𝑥,𝑦)|𝑥|𝑝+|𝑦|𝑝=1} with 𝑝1 satisfying the equation 1/𝑝+1/𝑝=1.

Similarly, if 𝑆 is an 𝑙2,𝑝-circle, 𝑝(0,1), corresponding to an antinorm (see Moszyńska and Richter [5]) then, according to Richter [2], the suitable arc-length measure is based upon the star disc 𝑆(𝑝)={(𝑥,𝑦)2|𝑥|𝑝+|𝑦|𝑝1} with 𝑝<0 satisfying 1/𝑝+1/𝑝=1. The star disc 𝑆(𝑝) corresponds to a specific semi-antinorm with respect to the canonical fan (see Moszyńska and Richter [5]).

The situation for ellipses has been discussed in Richter [3]. If 𝑆=𝐷𝑎,𝑏={(𝑥,𝑦)2(𝑥/𝑎)2+(𝑦/𝑏)21} is an elliptically contoured disc and 𝐸(𝑎,𝑏)=𝜕𝑆 its boundary then the suitable arc-length measure on the Borel 𝜎-field of subsets of the ellipse 𝐸(𝑎,𝑏) is based upon the disc 𝐷(1/𝑏,1/𝑎). Note that again 𝐷(𝑎,𝑏) and 𝐷(1/𝑏,1/𝑎) correspond to dual norms.

The arc-length measure used for defining the 𝑝-generalized circle number allows both for 𝑝1 and for 0<𝑝<1 the same additional interpretation in terms of the derivative of the area content function with respect to the 𝑝-radius variable. This way, the notion of the 𝑝-generalized circumference of the 𝑝-circle was introduced first in Richter [4] 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 [5] for proving such convergence if the 𝑝-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 𝐸(𝑎,𝑏). 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 𝑆-generalized circumference of an 𝑆-circle corresponding to a star body and will discuss in Section 2 both its general geometric meaning and its specific interpretation either when 𝑆 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.

In this sense, Section 2 deals mainly with generalized circle numbers for star bodies while Section 3 is devoted to a certain class of unbounded star discs.

Definition 1.1. Let 𝜆2 be the Lebesgue measure in 2,𝑆 a star body, and 𝐴𝑆(𝜚)=𝜆2(𝜚𝑆) the corresponding area content function. Then, 𝑑𝐴𝑑𝜚𝑆(𝜚)=𝔘𝑆(𝜚),𝜚>0,(1.1) will be called the 𝑆-generalized circumference of 𝜚𝑆 or the 𝑆-generalized arc-length of the boundary 𝑆(𝜚) of 𝜚𝑆, 𝜚𝑆={(𝜚𝑥,𝜚𝑦)2(𝑥,𝑦)𝑆}.

It follows from the properties of the Lebesgue measure that𝐴𝑆(𝜚)𝜚2=𝔘𝑆(𝜚)2𝜚=𝐴𝑆(1),𝜚>0.(1.2) The representation 𝐴𝑆(𝜚)=𝜚0𝔘𝑆(𝑟)𝑑𝑟(1.3)

may be understood as a generalized method of indivisibles for the Lebesgue measure, where the indivisibles are multiples of the boundary of 𝑆 and measuring their circumferences is based upon 𝔘𝑆. Equations (1.2) may suggest on the one hand to call 𝐴𝑆(1) the 𝑆-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 [14] 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 𝔘𝑆(𝜚). In other words, one naturally looks for a geometry such that the arc-length of 𝑆(𝜚) with respect to this geometry coincides with the 𝑆-generalized circumference 𝔘𝑆(𝜚). 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 [6] in connection with his fourth problem. If we can uniquely identify a geometry such that the arc-length measure of 𝑆, AL𝑆,𝑆(𝜚), which is based upon the geometry's unit ball 𝑆, satisfies AL𝑆,𝑆(𝜚)=𝔘𝑆(𝜚),(1.4) then we can observe already the nontrivial situation that𝐴𝑆(𝜚)𝜚2=AL𝑆,𝑆2𝜚=𝐴𝑆(1),𝜚>0.(1.5) At such stage of investigation, it will then be already much more motivated that the area content of the unit star, 𝐴𝑆(1), is called the 𝑆-generalized circle number, 𝜋(𝑆).

In this sense, the considerations in Richter [13] deal with restrictions of the function 𝑆𝜋(𝑆) to 𝑙2,𝑝-balls, 𝑝>0, and to axes aligned ellipses.

2. Star Bodies

A subset 𝑆 from 2 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 𝑧2 there exists a uniquely determined 𝜚>0 such that 𝑧/𝜚𝜕𝑆, where 𝜕𝑆 denotes the boundary of the set 𝑆. This 𝜚 equals the value of the Minkowski functional with respect to the reference set 𝑆𝑆(𝑥,𝑦)=inf{𝜆>0(𝑥,𝑦)𝜆𝑆},(2.1)

at any point (𝑥,𝑦)𝜕𝑆. The function 𝑆 is often called the gauge function of 𝑆 (see, e.g., in Webster [7]) and coincides, for (𝑥,𝑦)(0,0), with the reciprocal of the radial function (see, e.g., in Thompson [9] and Moszyńska [8]),𝜚𝑆((𝑥,𝑦))=sup{𝜆0𝜆(𝑥,𝑦)𝑆}.(2.2)

The special cases that 𝑆 is a norm or an antinorm are of particular interest and will be separately dealt with in Examples 2.13 and 2.14.

With a star body 𝑆, the pair (2,𝑆) may be considered as a generalized Minkowski plane. The star disc and the star circle of 𝑆-radius 𝜚 will be defined then by 𝐾𝑆(𝜚)={𝑟𝑠,𝑠𝑆,0𝑟𝜚} and 𝑆(𝜚)=𝜕𝐾𝑆(𝜚), respectively. The set 𝑆 will be called the unit star in this plane.

Let 𝑇 be another star disc in 2 which will be specified later. Whenever possible, we may define the 𝑇-arc-length of the curve 𝑆(𝜚) as follows.

Definition 2.1. If 𝑛={𝑧0,𝑧1,,𝑧𝑛=𝑧0} denotes a successive and positive (anticlockwise) oriented partition of 𝑆(𝜚), then the positive directed 𝑇-arc-length of 𝑆(𝜚) is defined by AL𝑆𝑇(𝜚)=lim𝑛𝑛𝑗=1𝑇𝑧𝑗𝑧𝑗1,(2.3) if the limit exists for and is independent of all described partitions of 𝑆(𝜚) with 𝐹(𝑛)=max1𝑗𝑛𝑇(𝑧𝑗𝑧𝑗1) tending to zero as 𝑛.

Using triangle inequality or its reverse, one can show that if 𝑆 is a norm or antinorm then taking the limit may be changed with taking the supremum or the infimum, respectively. Notice that because 𝑇 is in general not a symmetric function, the orientation in the partition may have essential influence onto the value of AL𝑆,𝑇(𝜚) and is, therefore, assumed here always to be positive.

For studying AL𝑆,𝑇(𝜚), let a parameter representation of the unit-𝑆-circle 𝑆(1) be given by 𝑆(1)={𝑅𝑆(𝜑)(cos𝜑,sin𝜑)𝑇,0𝜑<2𝜋}. Later on, we will assume that 𝑅𝑆 is a.e. differentiable. From the relation 𝑆(𝑥,𝑦)𝑇=1,(𝑥,𝑦)𝑇𝑆(1),(2.4)

it follows that 𝑆(cos𝜑,sin𝜑)𝑇=1𝑅𝑆(𝜑),0𝜑<2𝜋.(2.5)

In other words, with the notation 𝑀𝑆(𝜑)=𝑆((cos𝜑,sin𝜑)𝑇), we have𝑆(1)=cos𝜑𝑀𝑆,(𝜑)sin𝜑𝑀𝑆(𝜑)𝑇,0𝜑<2𝜋.(2.6)

This motivates the following definition which generalizes more particular notions from earlier considerations.

Definition 2.2. For an arbitrary star body 𝑆, the 𝑆-generalized sine and cosine functions are sin𝑆(𝜑)=sin𝜑𝑀𝑆(𝜑),cos𝑆(𝜑)=cos𝜑𝑀𝑆[(𝜑),𝜑0,2𝜋),(2.7) respectively.

Notice that there is an elementary geometric interpretation of these generalized trigonometric functions when one considers a right-angled triangl Tr=((0,0)𝑇,(𝑥,0)𝑇,(𝑥,𝑦)𝑇) with 𝑥>0 and 𝑦>0 as follows. The 𝑆-generalized sine and cosine of the angle 𝜑[0,2𝜋) between the directions of the positive 𝑥-axes and the line through the points (0,0)𝑇 and (𝑥,𝑦)𝑇 aresin𝑆𝑦(𝜑)=𝑆(𝑥,𝑦)𝑇,cos𝑆𝑥(𝜑)=𝑆(𝑥,𝑦)𝑇,(2.8)

respectively. These functions satisfy 𝑆cos𝑆(𝜑),sin𝑆(𝜑)=1,(2.9)

generalizing the well-known formula cos2𝜑+sin2𝜑=1.

Definition 2.3. The 𝑆-generalized polar coordinate transformation Pol𝑆[[0,)×0,2𝜋)2(2.10) is defined by 𝑥=𝑟cos𝑆(𝜑),𝑦=𝑟sin𝑆(𝜑),0𝜑<2𝜋,0𝑟<.(2.11)

Let us denote the quadrants in 2 in the usual anticlockwise ordering by 𝑄1 up to 𝑄4.

Theorem 2.4. The map Pol𝑆 is almost one-to-one, for 𝑥0, its inverse Pol1S is given by 𝑟=𝑆|||𝑦(𝑥,𝑦),arctan𝑥|||=𝜑𝑖𝑛𝑄1,𝜋𝜑𝑖𝑛𝑄2,𝜑𝜋𝑖𝑛𝑄3,2𝜋𝜑𝑖𝑛𝑄4,(2.12) and its Jacobian satisfies 𝐽(𝑟,𝜑)=𝐷(𝑥,𝑦)=𝑟𝐷(𝑟,𝜑)𝑀2𝑆.(𝜑)(2.13)

Proof. The proof follows that of Theorem 8 in Richter [3] and makes essentially use of the fact that the derivatives of the 𝑆-generalized trigonometric functions sin𝑆 and cos𝑆 allow the representations sin𝑆1(𝜑)=𝑀2𝑆(𝜑)cos𝜑𝑀𝑆(𝜑)sin𝜑𝑀𝑆,(𝜑)cos𝑆1(𝜑)=𝑀2𝑆(𝜑)sin𝜑𝑀𝑆(𝜑)cos𝜑𝑀𝑆.(𝜑)(2.14)

Using 𝑆-generalized polar coordinates, we can write𝑆(𝜚)=𝜚cos𝑆(𝜑),𝜚sin𝑆(𝜑)𝑇,0𝜑<2𝜋,𝜚>0.(2.15)

We assume from now on that 𝑇 is positively homogeneous, put𝑇𝑧𝑗𝑧𝑗1=𝑇Δ𝑗𝑥,Δ𝑗𝑦𝑇,(2.16)

and consider𝑇𝑧𝑗𝑧𝑗1=𝑇Δ𝑗𝑥(𝜑)Δ𝑗𝜑,Δ𝑗𝑦(𝜑)Δ𝑗𝜑𝑇Δ𝑗𝜑,(2.17)

for sufficiently thin partition 𝑛 and Δ𝑗𝜑>0. We get in the limit, which was assumed in Definition 2.1 to be uniquely determinedAL𝑆,𝑇(𝜚)=02𝜋𝑇𝑥(𝜑),𝑦(𝜑)𝑑𝜑=𝜚02𝜋𝑇𝑥(𝜑)𝜚,𝑦(𝜑)𝜚𝑑𝜑=𝜚02𝜋𝑇cos𝑆(𝜑),sin𝑆(𝜑)𝑇𝑑𝜑=𝜚AL𝑆,𝑇(1).(2.18)

It follows from the proof of Theorem 2.4 thatAL𝑆,𝑇(𝜚)=𝜚02𝜋𝑅2𝑆(𝜑)𝑇𝑀𝑆(𝜑)sin𝜑cos𝜑+𝑀𝑆(𝜑)cos𝜑sin𝜑𝑑𝜑,(2.19)

that is,AL𝑆,𝑇(𝜚)=𝜚02𝜋𝑅2𝑆(𝜑)𝑇𝑂(𝜑)𝔵𝑆(𝜑)𝑑𝜑,(2.20) with𝑂(𝜑)=cos𝜑sin𝜑sin𝜑cos𝜑,𝔵𝑆𝑅(𝜑)=𝑆(𝜑)𝑅2𝑆1(𝜑)𝑅𝑆(𝜑).(2.21)

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 𝑆,𝑥 and 𝑆,𝑦 of the function (𝑥,𝑦)𝑆(𝑥,𝑦) satisfy the representation 0110𝑆,𝑥𝑟cos𝑠(𝜑),𝑟sin𝑠(𝜑)𝑆,𝑦𝑟cos𝑠(𝜑),𝑟sin𝑠(𝜑)=𝔒(𝜑)𝔵𝑆(𝜑).(2.22)

Proof. It follows from the relation 𝑆𝑟cos𝑆(𝜑),𝑟sin𝑆(𝜑)=𝑟(2.23) that the partial derivatives 𝑆,𝑥 and 𝑆,𝑦 satisfy the equation system 𝜕𝜕𝜑𝑆cos𝑆(𝜑),sin𝑆𝜕(𝜑)=0,𝜕𝑟𝑆𝑟cos𝑆(𝜑),𝑟sin𝑆(𝜑)=1.(2.24) Solving this differential equation system, we get 𝑆,𝑥𝑟cos𝑠(𝜑),𝑟sin𝑠=1(𝜑)𝑅2𝑆𝑅(𝜑)𝑆(𝜑)cos(𝜑)+𝑅𝑆(𝜑)sin(𝜑),𝑆,𝑦𝑟cos𝑠(𝜑),𝑟sin𝑠=1(𝜑)𝑅2𝑆𝑅(𝜑)𝑆(𝜑)sin(𝜑)𝑅𝑆(𝜑)cos(𝜑).(2.25) Hence, 𝑆,𝑥𝑟cos𝑠(𝜑),𝑟sin𝑠(𝜑)𝑆,𝑦𝑟cos𝑠(𝜑),𝑟sin𝑠=(𝜑)0110𝔒(𝜑)𝔵𝑆(𝜑).(2.26)

Let 𝐵 be a 2×2-matrix and 𝐵𝑇={𝐵(𝑥,𝑦)𝑇(𝑥,𝑦)𝑇𝑇}. Clearly, multiplying the set 𝑇 by the matrix 0110 causes an anticlockwise rotation of 𝑇 through the angle 𝜋/2. Hence, if 𝑇 is a star disc, then 𝐵𝑇 is a star disc too.

Corollary 2.6. For positively homogeneous 𝑇, differentiable 𝑆, formula (2.20) may be rewritten as AL𝑆,𝑇(𝜚)=𝜚02𝜋𝑅2𝑆(𝜑)𝑇0110𝑆(𝑥,𝑦)|(𝑥,𝑦)=Pol𝑆(𝑟,𝜑)𝑑𝜑.(2.27)

Proof. Based upon Lemma 2.5, formula (2.20) may be reformulated as AL𝑆,𝑇(𝜚)=𝜚02𝜋𝑅2𝑆(𝜑)𝑇0110𝑆,𝑥𝑟cos𝑠(𝜑),𝑟sin𝑠(𝜑)𝑆,𝑦𝑟cos𝑠(𝜑),𝑟sin𝑠(𝜑)𝑑𝜑.(2.28) Because of 𝑇𝜉𝜂𝜉𝜂𝜉𝜂𝑇0110=inf𝜆>00110𝜆𝑇=inf𝜆>0𝜆0110=𝑇0110𝜉𝜂,(2.29) it follows the assertion.

Remark 2.7. The plug-in version 𝑆(𝑥,𝑦)|(𝑥,𝑦)=Pol𝑆(𝑟,𝜑) of the gradient 𝑆(𝑥,𝑦) coincides with the image of the gradient 𝑆(𝑥,𝑦) after changing Cartesian with 𝑆-generalized polar coordinates, Pol𝑆(𝑆(𝑥,𝑦))(𝑟,𝜑).

Proof. Changing Cartesian coordinates (𝑥,𝑦) with 𝑆-generalized polar coordinates (𝑟,𝜑), we have 𝑟=𝑆(𝑥,𝑦) and 𝜑=arctan(𝑦/𝑥). Starting from 𝜕𝜕𝑥𝑆𝜕(𝑥,𝑦)=𝜕𝑟𝑆𝜕(𝑥,𝑦)𝜕𝜕𝑥𝑟+𝜕𝜑𝑆𝜕(𝑥,𝑦)𝜕𝑥𝜑,(2.30) and the analogous one for (𝜕/𝜕𝑦)𝑆(𝑥,𝑦), and taking into account that 𝜕𝜕𝑟𝑆(𝑥,𝑦)|(𝑥,𝑦)=Pol𝑆(𝑟,𝜑)=𝜕𝜕𝑟𝑆𝑟cos𝑆(𝜑),𝑟sin𝑆𝜕(𝜑)=1,𝜕𝜑𝑆(𝑥,𝑦)|(𝑥,𝑦)=Pol𝑆(𝑟,𝜑)=𝜕𝜕𝜑𝑆𝑟cos𝑆(𝜑),𝑟sin𝑆(𝜑)=0,(2.31) it follows 𝜕𝜕𝑥𝑆(𝑥,𝑦)=𝜕𝑟|𝜕𝑥(𝑥,𝑦)=Pol𝑆(𝑟,𝜑)=𝑆,𝑥𝑟cos𝑆(𝜑),𝑟sin𝑆,𝜕(𝜑)𝜕𝑦𝑆(𝑥,𝑦)=𝜕𝑟|𝜕𝑦(𝑥,𝑦)=Pol𝑆(𝑟,𝜑)=𝑆,𝑦𝑟cos𝑆(𝜑),𝑟sin𝑆.(𝜑)(2.32)

Remark 2.8. For positively homogeneous 𝑇 and differentiable Minkowski functional 𝑆 of the star disc 𝑆, formula (2.20) may be rewritten as AL𝑆,𝑇(𝜚)=𝜚02𝜋𝑅2𝑆(𝜑)𝑇0110Pol𝑆𝑆(𝑥,𝑦)(𝑟,𝜑)𝑑𝜑.(2.33)

Definition 2.9. A star body 𝑆 and a star disc 𝑇 satisfy the rotated gradient condition if 𝑇0110𝑆(𝑥,𝑦)|(𝑥,𝑦)=Pol𝑆(𝑟,𝜑)=1,a.e.(2.34)

Let us notice that at the point (𝑥,𝑦) from 𝑆(𝜚), the gradient 𝑆(𝑥,𝑦) is normal to the level set 𝑆(𝜚),𝜚>0 of 𝑆. The following lemma is a consequence of the above consideration.

Lemma 2.10. For a star body 𝑆 and a star disc 𝑆 satisfying the rotated gradient condition (2.34), the positive directed 𝑆-arc-length of 𝑆(𝜚) allows the representation ALS,𝑆(𝜚)=𝜚02𝜋𝑅2𝑠(𝜑)𝑑𝜑.(2.35)

We consider now the area function𝐴𝑆(𝜚)=𝜚2𝐴𝑆𝜚(1)=2202𝜋𝑅2𝑆(𝜑)𝑑𝜑,(2.36)

where 𝐴𝑆(1) denotes the area content of the unit disc 𝐾𝑆(1). The derivative of the area function satisfies obviously 𝑑𝐴𝑑𝜚𝑆(𝜚)=𝜚02𝜋𝑅2𝑆(𝜑)𝑑𝜑.(2.37)

The following theorem has thus been proved.

Theorem 2.11. If the star body 𝑆 and the star disc 𝑆 satisfy the rotated gradient condition (2.34), then 𝔘𝑆(𝜚)=AL𝑆,𝑆(𝜚),(2.38) that is, the 𝑆-generalized circumference of 𝑆 coincides with the positive directed 𝑆-circumference of 𝑆.

If relation (2.38) holds, thenAL𝑆,𝑆(1)=2𝐴𝑆(1).(2.39) Consequently, the ratios 𝐴𝑆(𝜚)/𝜚2 and AL𝑆,𝑆(𝜚)/2𝜚 satisfy the relations𝐴𝑆(𝜚)𝜚2(a)=𝐴𝑆(1)(c)=AL𝑆,𝑆(𝜚)2𝜚,𝜚>0.(2.40) In this sense, the geometry and the arc-length measure generated by 𝑆 fulfill our expectations. The following definition is thus well motivated if a star body 𝑆 and a star disc 𝑆 are chosen in such a way that the limit in Definition 2.1 is uniquely determined, 𝑇 is positively homogeneous, 𝑆 is a.e. differentiable, and the rotated gradient condition (2.34) is satisfied.

Definition 2.12. (a) The properties of the star bodies 𝐾𝑆(𝜚),𝜚>0, which are expressed by the equations (a) and (c) in (2.40) are called the area content and the 𝑆-generalized circumference properties of the discs, respectively.
(b) The quantity 𝐴𝑆(1)=𝜋(𝑆) is called the 𝑆-generalized circle number of the star bodies 𝐾𝑆(𝜚),𝜚>0.

We may write now (2.40) asAL𝑆,𝑆(𝜚)=2𝜋(𝑆)𝜚,𝐴𝑆(𝜚)=𝜋(𝑆)𝜚2.(2.41) Notice that the circle number function 𝑆𝜋(𝑆) assigns a generalized circle number to any star body 𝐾𝑆(𝜚) satisfying assumption (2.34). The restrictions of this function to 𝑙2,𝑝-balls or axes aligned ellipses were considered in Richter [13].

Example 2.13. Here, we consider a first, rather general case, where the rotated gradient condition (2.34) is satisfied. Let (𝑝) and (𝑑) denote a (primary) 𝐶1-norm in 𝑛 and the corresponding dual one, respectively. It is proved in Yang [10] that 𝔵(𝑝)(𝑑)=1,𝔵𝑛.(2.42) Hence, if 𝑆 is a convex body, that is, 𝑆(𝔵)=𝔵(𝑝) is a (primary) norm, and if 0110𝑇=𝔵𝑛𝔵(𝑑)1=𝑆(2.43)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 𝜋(𝑆)=𝐴𝑆(1) we may refer, for example, to Pisier [11], where volumes of convex bodies are dealt with. Alternatively, one may use 𝑆-generalized polar coordinates for making the respective calculations in given cases. The particular results for 𝑙2,𝑝-circles with 𝑝1 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 𝑙2,𝑝-circles 𝐶𝑝={(𝑥,𝑦)2|𝑥|𝑝+|𝑦|𝑝=1}  with  0<𝑝<1. Such generalized circles correspond to antinorms. A suitable arc-length measure for measuring 𝐶𝑝 is based upon the star disc 𝑆(𝑝)={(𝑥,𝑦)2|𝑥|𝑝+|𝑦|𝑝1}  with  𝑝<0 satisfying 1/𝑝+1/𝑝=1. The star discs 𝑆(𝑝)  are closely related to specific semiantinorms with respect to the canonical fan. The corresponding generalized circle numbers have been determined in Richter [2]. 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 𝑇 satisfying condition (2.34) for sets 𝑆 being generated by antinorms.

As was indicated in Richter [2], 𝑝-generalized circle numbers for 0<𝑝<1 may occur, for example, within certain combinatorial formulae. Notice further that the reciprocal values of the coefficients of the binomial series expansion14=14𝑢𝑛=01𝜋(1/𝑛)𝑢𝑛1,𝑢0,4,(2.44)

are just the generalized circle numbers corresponding to the nonconvex 𝑙2,1/𝑛-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 IR2 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 𝑝<0, the function||||(𝑥,𝑦)(𝑥,𝑦)𝑝=|𝑥|𝑝+||𝑦||𝑝1/𝑝,𝑥0,𝑦0,(3.1)

which denotes a semi-antinorm, and the 𝑝-generalized circle𝐶𝑝=(𝑥,𝑦)2||||(𝑥,𝑦)𝑝=1.(3.2)

The pairs of straight lines |𝑦|=1 and |𝑥|=1 represent asymptotes for the circle 𝐶𝑝 as |𝑥| or |𝑦|, respectively. The intersection point of the 𝑝-circle 𝐶𝑝 with the line 𝑦=𝑥 is for positive coordinates (𝑥0,𝑦0)=21/𝑝(1,1).

Let further 𝐶𝑝(𝑟)=𝑟𝐶𝑝,𝑟>0 denote the 𝑝-generalized circle of 𝑝-generalized radius 𝑟>0. It is the boundary of the unbounded 𝑝-generalized disc of 𝑝-generalized radius 𝑟,𝐾𝑝(𝑟)=(𝑥,𝑦)𝑅2||||(𝑥,𝑦)𝑝𝑟=𝑟𝐾𝑝,𝐾𝑝=𝐾𝑝(1).(3.3)

As because||||(𝑥,𝑦)𝑝𝑟|𝑥|𝑝+||𝑦||𝑝𝑟𝑝||𝑦||𝑓(|𝑥|)+𝑓𝑓(𝑟)for𝑓(𝜆)=𝜆𝑝,𝜆0,(3.4)

one may call 𝐾𝑝(𝑟) a two-dimensional Orlicz antiball corresponding to the Young function 𝑓. The disc 𝐾𝑝 is a star-shaped but noncompact set and, therefore, not a star body. For any (𝑥,𝑦)𝐶𝑝(𝑟) one may think of 𝑟 as the value of the Minkowski functional with respect to the reference set 𝐾𝑝. The area content and the Euclidean circumference of the unit 𝑝-circle are obviously unbounded. That is why we consider from now on truncated 𝑝-circles. To this end, let us introduce truncation cones𝐶𝑥1=(𝑥,𝑦)2(𝑥,𝑦)Π1(𝑥,𝑦)Π1<𝑥(𝑥,𝑦)1,𝑦1Π1𝑥1,𝑦1Π1𝑥1,𝑦1=(𝑥,𝑦)2||||𝑥𝑦||||<||𝑥𝑥+𝑦1𝑦1||||𝑥1+𝑦1||,(3.5)

where denotes Euclidean norm, 1=(1,1), 𝑥1 is chosen according to 𝑥1>𝑥0=21/𝑝 and |𝑦1|=(1|𝑥1|𝑝)1/𝑝<1.

The question of interest is now whether we may define in a reasonable way circle numbers for the truncated 𝑝-discs 𝐾𝑥1𝑝(𝑟)=𝑟𝐾𝑥1𝑝, the boundaries of which are the 𝑝-circles 𝐶𝑥1𝑝(𝑟)=𝑟𝐶𝑥1𝑝 of 𝑝-radius 𝑟, where𝐾𝑥1𝑝=𝐾𝑝𝑥𝐶1,𝐶𝑥1𝑝=𝐶𝑝𝑥𝐶1.(3.6)

To this end, let 𝑛=(𝑧0,𝑧1,,𝑧𝑛) be an arbitrary successive anticlockwise-oriented partition of the truncated circle 𝐶𝑥1𝑝 satisfying 𝑧0=(𝑥0,(1𝑥𝑝0)1/𝑝) and 𝑧𝑛=(𝑥1,(1𝑥𝑝1)1/𝑝). We consider the sum𝑆𝑛=𝑛𝑗=1||𝑧𝑗𝑧𝑗1||𝑞=𝑛𝑗=1||𝑥𝑗𝑥𝑗1||𝑞+||𝑦𝑗𝑦𝑗1||𝑞1/𝑞,𝑞(0,1),(3.7)

and observe that due to the reverse triangle inequality it decreases monotonously as𝐹𝑛=sup1𝑗𝑛||𝑧𝑗𝑧𝑗1||𝑞0.(3.8)

According to the symmetry of 𝐶𝑥1𝑝, the following remark is justified.

Remark 3.1. For 𝑞(0,1), the 𝑙2,𝑞-arc-length of the truncated circle 𝐶𝑥1𝑝 is defined as AL𝑥1𝑝,𝑞=8lim𝐹𝑛0𝑆𝑛.(3.9)

If 𝑥𝑦(𝑥) denotes an arbitrary parameter representation of the truncated circle 𝐶𝑥1𝑝, then18AL𝑥1𝑝,𝑞=lim𝐹𝑛𝑛0𝑗=1||||1+Δ𝑦𝑗Δ𝑥𝑗||||𝑞1/𝑞Δ𝑥𝑗=𝑥1𝑥0||𝑦1+||(𝑥)𝑞1/𝑞𝑑𝑥.(3.10)

Let us denote the usual Euclidean area content of the truncated circle disc 𝐾𝑥1𝑝 by 𝐴𝑥1𝑝.

Lemma 3.2. Let for arbitrary 𝑝<0 the number 𝑝(0,1) be uniquely defined by the equation 1/𝑝+1/𝑝=1. Then, 𝐴𝑥1𝑝=12AL𝑥1𝑝,𝑝.(3.11)

Proof. With 𝑦(𝑥)=(1|𝑥|𝑝)1/𝑝=(1𝑥𝑝)1/𝑝,𝑦(𝑥)=(1𝑥𝑝)1/𝑝1𝑥𝑝1,(3.12) it follows from the above formulae that AL𝑥1𝑝,𝑞=8𝑥11/21/𝑝1+(1𝑥𝑝)(1/𝑝1)𝑞𝑥(𝑝1)𝑞1/𝑞𝑑𝑥.(3.13) Changing variables 𝑢=𝑥𝑝,𝑑𝑥=𝑑𝑢/(𝑝𝑢(𝑝1)/𝑝) causes a change of the limits of integration: AL𝑥1𝑝,𝑞=8𝑝𝑥1/2𝑝11+(1𝑢)((1𝑝)/𝑝)𝑞𝑢((𝑝1)/𝑝)𝑞1/𝑞𝑑𝑢𝑢(𝑝1)/𝑝=8||𝑝||𝑥1/2𝑝1𝑢((1𝑝)/𝑝)𝑞+(1𝑢)((1𝑝)/𝑝)𝑞1/𝑞𝑑𝑢.(3.14) Assuming now 1/𝑝+1/𝑞=1, or equivalently 𝑞=𝑝/(𝑝1)=𝑝, it follows that AL𝑥1𝑝,𝑝=8||𝑝||𝑥1/2𝑝1𝑢1+(1𝑢)1(𝑝1)/𝑝8𝑑𝑢=||𝑝||𝑥1/2𝑝11𝑢+𝑢𝑢(1𝑢)11/𝑝𝑑𝑢.(3.15) Hence, AL𝑥1𝑝,𝑝=8||𝑝||𝑥1/2𝑝1𝑢1/𝑝1(1𝑢)1/𝑝1𝑑𝑢.(3.16) Now, what about the area content of the truncated circle 𝐶𝑥1𝑝? The 𝑙2,𝑝-generalized standard triangle coordinate transformation Tr from Richter [4] is defined by Tr𝑝||𝜇||(𝑟,𝜇)=(𝑥,𝑦)with𝑥=𝑟𝜇,𝑦=+()𝑟1𝑝1/𝑝.(3.17) Because of (𝑥,𝑦)|𝑥|𝑝+||𝑦||𝑝=1,𝑥0𝑥𝑥1=Tr𝑝𝑥{1}×0,𝑥1,𝑟(𝑥,𝑦)|𝑥|𝑝+||𝑦||𝑝=1,𝑥0𝑥𝑥1=(𝑟𝑥,𝑟𝑦)|𝑥|𝑝+||𝑦||𝑝=1,𝑥0𝑥𝑥1=|||𝜉(𝜉,𝜂)𝑟|||𝑝+|||𝜂𝑟|||𝑝=1,𝑥0𝜉𝑟𝑥1=(𝑥,𝑦)|𝑥|𝑝+||𝑦||𝑝=𝑟𝑝,𝑥0𝑥𝑟(=𝜇)𝑥1=Tr𝑝𝑥{𝑟}×0,𝑥1,(3.18) it follows 0𝑟1𝑟(𝑥,𝑦)|𝑥|𝑝+||𝑦||𝑝=1,𝑥0𝑥𝑥1=Tr𝑝[]×𝑥0,10,𝑥1=𝐾𝑥1𝑝.(3.19) That is, 𝐾𝑥1𝑝=Tr𝑝[]×𝑥0,10,𝑥1.(3.20) Changing Cartesian with standard triangle coordinates in the integral 𝐴𝑥1𝑝=𝐾𝑥1𝑝𝑑(𝑥,𝑦),(3.21) we get 𝐴𝑥1𝑝=81𝑟=0𝑥1𝜇=𝑥0𝑟(1𝜇𝑝)(1𝑝)/𝑝8𝑑𝜇𝑑𝑟=2𝑥121/𝑝(1𝜇𝑝)1/𝑝1𝑑𝜇.(3.22) Substituting 𝑦=𝜇𝑝,𝑑𝑦/𝑑𝜇=𝑝𝑦(𝑝1)/𝑝, it follows that 𝐴𝑥1𝑝=4𝑝𝑥𝑝11/2(1𝑦)1/𝑝1𝑦1/𝑝14𝑑𝑦=||𝑝||𝑥1/2𝑝1𝑦1/𝑝1(1𝑦)1/𝑝1𝑑𝑦.(3.23) Hence, the lemma is proved.

Remark 3.3. Because of 𝑝<0, 𝐴𝑥1𝑝as𝑥1.(3.24)

The following corollary and definition are now quite obvious and well motivated.

Corollary 3.4. For arbitrary 𝑥1>𝑥0=21/𝑝, the truncated star discs 𝐾𝑥1𝑝 have the area content and 𝑝-generalized circumference properties (𝑎) and (𝑐), respectively, 𝐴𝑥1𝑝(𝑟)𝑟2𝑎=𝐴𝑥1𝑝(𝑐)=AL𝑥1𝑝,𝑝(𝑟),2𝑟(3.25) from which it follows immediately 𝑑𝐴𝑑𝑟𝑥1𝑝(𝑟)=AL𝑥1𝑝,𝑝(𝑟).(3.26)

Remark 3.5. One may think of (3.26) as reflecting a generalized method of indivisibles for each 𝑥1>𝑥0, where the truncated circles 𝐶𝑥1𝑝 are the indivisibles and measuring them is based upon the geometry generated by the disc 𝐾𝑝.

Definition 3.6. For arbitrary 𝑥1>21/𝑝, the quantity 𝐴𝑥1𝑝=𝜋𝑥1(𝑝) will be called the circle number of the truncated circle 𝐶𝑥1𝑝.