Circle Numbers for Star Discs

The notion of a generalized circle number which has recently been discussed for l 2 ,p -circles and ellipses will be extended here for star bodies and a class of unbounded star discs.


Introduction
Generalized circle numbers have been discussed for l 2,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  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  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 2 ISRN Geometry 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 IR 2 , and let its area content be defined as usual by its Lebesgue measure. Furthermore, let us call the boundary of times the star disc S the S-circle of S-radius , > 0 and denote it by S . 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-Sradius 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 > 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 l 2,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 l 2,p * -circle { x, y : |x| p * |y| p * 1} with p * ≥ 1 satisfying the equation 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 x/a 2 y/b 2 ≤ 1} 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 p ≥ 1 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 λ 2 be the Lebesgue measure in Ê 2 , S a star body, and A S λ 2 S the corresponding area content function. Then, 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 S and measuring their circumferences is based upon Í S . Equations 1.2 may suggest on the one hand to call A S 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  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 Í S . In other words, one naturally looks for a geometry such that the arc-length of S with respect to this geometry coincides with the Sgeneralized circumference Í S . 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, AL S,S * , which is based upon the geometry's unit ball S * , satisfies ISRN Geometry 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, A S 1 , is called the S-generalized circle number, π S . In this sense, the considerations in Richter 1-3 deal with restrictions of the function S → π S to l 2,p -balls, p > 0, and to axes aligned ellipses.

Star Bodies
A subset S from Ê 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 z ∈ Ê 2 there exists a uniquely determined > 0 such that z/ ∈ ∂S, where ∂S denotes the boundary of the set S. This equals the value of the Minkowski functional with respect to the reference set S The function h 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 , The special cases that h 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 Ê 2 , h S may be considered as a generalized Minkowski plane. The star disc and the star circle of S-radius will be defined then by K S {r · s, s ∈ S, 0 ≤ r ≤ } and S ∂K S , respectively. The set S will be called the unit star in this plane.
Let T be another star disc in Ê 2 which will be specified later. Whenever possible, we may define the T-arc-length of the curve S as follows.
if the limit exists for and is independent of all described partitions of S with F n max 1≤j≤n h T z j − z j−1 tending to zero as n → ∞.
Using triangle inequality or its reverse, one can show that if h S is a norm or antinorm then taking the limit may be changed with taking the supremum or the infimum, respectively.

5
Notice that because h T is in general not a symmetric function, the orientation in the partition may have essential influence onto the value of AL S,T and is, therefore, assumed here always to be positive.
For studying AL S,T , let a parameter representation of the unit-S-circle S 1 be given by S 1 {R S ϕ cos ϕ, sin ϕ T , 0 ≤ ϕ < 2π}. Later on, we will assume that R S is a.e. differentiable. From the relation In other words, with the notation M S ϕ h S cos ϕ, sin ϕ T , we have This motivates the following definition which generalizes more particular notions from earlier considerations. 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 are is defined by x r cos S ϕ , y r sin S ϕ , 0 ≤ ϕ < 2π, 0 ≤ r < ∞.

2.11
Let us denote the quadrants in Ê 2 in the usual anticlockwise ordering by Q 1 up to Q 4 .

Theorem 2.4. The map
Pol S is almost one-to-one, for x / 0, its inverse Pol −1 S is given by

2.12
and its Jacobian satisfies

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 sin S and cos S allow the representations 2.14 Using S-generalized polar coordinates, we can write S cos S ϕ , sin S ϕ T , 0 ≤ ϕ < 2π, > 0 .

2.15
We assume from now on that h T is positively homogeneous, put and consider for sufficiently thin partition n and Δ j ϕ > 0. We get in the limit, which was assumed in Definition 2.1 to be uniquely determined

2.18
It follows from the proof of Theorem 2.4 that 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 h S,x and h S,y of the function x, y → h S x, y satisfy the representation

2.26
Let B be a 2 × 2-matrix and BT {B x, y T : x, y T ∈ T}. Clearly, multiplying the set T by the matrix 0 −1 1 0 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 h T , differentiable h S , formula 2.20 may be rewritten as
Proof. Based upon Lemma 2.5, formula 2.20 may be reformulated as h S,x r cos s ϕ , r sin s ϕ h S,y r cos s ϕ , r sin s ϕ dϕ.

2.28
Because of

2.29
it follows the assertion.
Remark 2.7. The plug-in version ∇h S x, y | x,y Pol S r,ϕ of the gradient ∇h S x, y coincides with the image of the gradient ∇h S x, y after changing Cartesian with S-generalized polar coordinates, Pol S ∇h S x, y r, ϕ . Let us notice that at the point x, y from S , the gradient ∇h S x, y is normal to the level set S , > 0 of h S . The following lemma is a consequence of the above consideration.

2.35
We consider now the area function where A S 1 denotes the area content of the unit disc K S 1 . The derivative of the area function satisfies obviously The following theorem has thus been proved. 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, h T is positively homogeneous, h S is a.e. differentiable, and the rotated gradient condition 2.34 is satisfied.
Definition 2.12. a The properties of the star bodies K S , > 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 A S 1 : π S is called the S-generalized circle number of the star bodies K S , > 0.
We may write now 2.40 as AL S,S * 2π S , A S π S 2 .

2.41
Notice that the circle number function S → π S assigns a generalized circle number to any star body K S satisfying assumption 2.34 . The restrictions of this function to l 2,p -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 · p and · d denote a primary C 1 -norm in Ê n and the corresponding dual one, respectively. It is proved in Yang 10 that

2.42
Hence, if S is a convex body, that is, h S Ü Ü p is a primary norm, and if 0 1 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 A S 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 l 2,p -circles with p ≥ 1 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 l 2,p -circles C p { x, y ∈ Ê 2 : |x| p |y| p 1} with 0 < p < 1. Such generalized circles correspond to antinorms. A suitable arc-length measure for measuring C p is based upon the star disc S p * * { x, y ∈ Ê 2 : |x| p * * |y| p * * ≥ 1} 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 expansion are just the generalized circle numbers corresponding to the nonconvex l 2,1/n -circles. One could also ask for a possibly elementary geometric? explanation of this fact.

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 IR 2 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 p 1/p , x / 0, y / 0, 3.1 which denotes a semi-antinorm, and the p-generalized circle C p x, y ∈ Ê 2 : x, y p 1 .

3.2
The pairs of straight lines |y| 1 and |x| 1 represent asymptotes for the circle C p as |x| → ∞ or |y| → ∞, respectively. The intersection point of the p-circle C p with the line y x is for positive coordinates x 0 , y 0 2 −1/p · 1, 1 . Let further C p r r · C p , 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, K p r x, y ∈ R 2 : x, y p ≤ r rK p , K p K p 1 .

3.3
As because x, y p ≤ r ⇐⇒ |x| p y p ≥ r p ⇐⇒ f |x| f y ≥ f r for f λ λ p , λ ≥ 0, 3.4 one may call K p r a two-dimensional Orlicz antiball corresponding to the Young function f. The disc K p is a star-shaped but noncompact set and, therefore, not a star body. For any x, y ∈ C p r one may think of r as the value of the Minkowski functional with respect to the reference set K p . 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 cones C x 1 : x, y ∈ Ê 2 : x, y − Π 1 x, y Π 1 x, y < x 1 , y 1 − Π 1 x 1 , y 1 Π 1 x 1 , y 1 x, y ∈ Ê 2 : x − y x y < x 1 − y 1 x 1 y 1 ,