Research Article Circle Numbers for Star Discs

International Scholarly Research Network ISRN Geometry Volume 211, Article ID 479262, 16 ages doi:1.542/211/479262 Research Article Circle Numbers for Star Discs W.-D. Richter Institute of Mathematics, University of Rostock, Ulmenstraße 69, Haus 3, 1857 Rostock, Germany Corresondence should be addressed to W.-D. Richter, wolf-dieter.richter@uni-rostock.de Received 2 March 211; Acceted 8 Aril 211 Academic Editors: A. Belhaj, A. Borowiec, and A. Fino Coyright q 211 W.-D. Richter. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. The notion of a generalized circle number which has recently been discussed for l 2, -circles and ellises will be extended here for star bodies and a class of unbounded star discs. 1. Introduction Generalized circle numbers have been discussed for l 2, -circles in Richter 1, 2 and for ellises in Richter 3. All these numbers corresond on the one hand to an area content roerty of the considered discs which is based uon the usual, that is, Euclidean, area content measure and a suitably adoted radius variable. On the other hand, they reflect a circumference roerty of the considered generalized circles with resect to a non-euclidean length measure which is generated by a suitably chosen non-euclidean disc. Several basic and secific roerties 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 resect to the adoted 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 resect to the adoted 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 uon 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 alications 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 resent aer, we will rove that this method still alies when generalized circle numbers are derived for general

2 ISRN Geometry star discs. In this sense, this aer deals with bounded and unbounded star discs. Notice that because we will not assume symmetry of the unit disc, distances will deend on directions in general. To become more secific, 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 ϱ times the star disc S the S-circle of S-radius ϱ, ϱ>and denote it by C S ϱ. If we define the erimeter of S by using different length measures, then we can observe different erimeter-to- two-times- S-radius ratios, and these ratios differ from the corresonding area-content -to- squared-sradius ratio in general. If we choose, however, the length measure in a certain secific way, then the first ratio coincides with the second one for all ϱ>. In the most famous case when S is the Euclidean disc and measuring circumference is based uon 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, -circle, centered at the origin and thus defining a norm then, according to Richter 1, the suitable arc-length measure is based uon the dual norm, that is, the norm which is generated by the l 2, -circle { x, y : x y 1} with 1 satisfying the equation 1/ 1/ 1. Similarly, if S is an l 2, -circle,, 1, corresonding to an antinorm see Moszyńska and Richter 5 then, according to Richter 2, the suitable arc-length measure is based uon the star disc S { x, y R 2 : x y 1} with < satisfying 1/ 1/ 1. The star disc S corresonds to a secific semi-antinorm with resect to the canonical fan see Moszyńska and Richter 5. The situation for ellises has been discussed in Richter 3. IfS D a,b { x, y R 2 : x/a 2 y/b 2 1} is an ellitically contoured disc and E a,b S its boundary then the suitable arc-length measure on the Borel σ-field of subsets of the ellise E a,b is based uon the disc D 1/b,1/a. Note that again D a,b and D 1/b,1/a corresond to dual norms. The arc-length measure used for defining the -generalized circle number allows both for 1andfor<<1the same additional interretation in terms of the derivative of the area content function with resect to the -radius variable. This way, the notion of the -generalized circumference of the -circle was introduced first in Richter 4 under more general circumstances and motivated there by several of its alications. Several geometric interretations of this notion in cases of secial 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 rove the so-called thin layers roerty 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 isoerimetric roblem. From a technical oint 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 roving such convergence if the -generalized circle corresonds to an antinorm. The arc-length measure used for defining ellises numbers has also an interretation in terms of the derivative of the area content function but with resect to a generalized radius variable corresonding 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 alications, for examle, from robability 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 corresonding to a star body and will discuss in Section 2 both its general

ISRN Geometry 3 geometric meaning and its secific interretation either when S is generated by an arbitrary norm or by an antinorm of secial tye. 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 R 2,S astarbody,anda S ϱ λ 2 ϱs the corresonding area content function. Then, d dϱ A ( ) ( ) S ϱ : US ϱ, ϱ >, 1.1 will be called the S-generalized circumference of ϱs or the S-generalized arc-length of the boundary C S ϱ of ϱs, ϱs { ϱx, ϱy R 2 : x, y S}. It follows from the roerties of the Lebesgue measure that A S ) ϱ 2 ( ) U S ϱ A S 1, ϱ >. 1.2 2ϱ The reresentation ( ) ϱ A S ϱ U S r dr 1.3 may be understood as a generalized method of indivisibles for the Lebesgue measure, where the indivisibles are multiles of the boundary of S and measuring their circumferences is based uon U 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 aers of Richter 1 4 which are closely connected with this aroach allow a new look onto a class of geometric measure reresentations or, similarly, onto a class of stochastic reresentations which are quite fruitful for many alications. Several of these alications, esecially in robability theory and mathematical statistics, are discussed therein. Clearly, there is always a necessity to give a geometric or otherwise mathematical interretation of the circumference U S ϱ. In other words, one naturally looks for a geometry such that the arc-length of C S ϱ with resect to this geometry coincides with the S- generalized circumference U S 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 roblem. If we can uniquely identify a geometry such that the arc-length measure of S,AL S,S ϱ, which is based uon the geometry s unit ball S,satisfies AL S,S ) US ), 1.4

4 ISRN Geometry then we can observe already the nontrivial situation that ( ) A S ϱ ALS,S A S 1, ϱ >. 1.5 2ϱ ϱ 2 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, -balls, >, and to axes aligned ellises. 2. Star Bodies AsubsetS from R 2 is called a star body if it is star-shaed with resect to the origin and comact and has the origin in its interior. A set of this tye has the roerty that for every z R 2 there exists a uniquely determined ϱ>suchthatz/ϱ S, where S denotes the boundary of the set S. This ϱ equals the value of the Minkowski functional with resect to the reference set S h S ( x, y ) inf { λ>: ( x, y ) λs }, 2.1 at any oint x, y S. The function h S is often called the gauge function of S see, e.g., in Webster 7 and coincides, for x, y /,, with the recirocal of the radial function see, e.g., in Thomson 9 and Moszyńska 8, ϱ S (( x, y )) su { λ :λ ( x, y ) S }. 2.2 The secial cases that h S is a norm or an antinorm are of articular interest and will be searately dealt with in Examles 2.13 and 2.14. With a star body S, theair R 2,h S may be considered as a generalized Minkowski lane. The star disc and the star circle of S-radius ϱ will be defined then by K S ϱ {r s, s S, r ϱ} and C S ϱ K S ϱ, resectively. The set S will be called the unit star in this lane. Let T beanotherstardiscinr 2 which will be secified later. Whenever ossible, we may define the T-arc-length of the curve C S ϱ as follows. Definition 2.1. If Z n {z,z 1,...,z n z } denotes a successive and ositive anticlockwise oriented artition of C S ϱ, then the ositive directed T-arc-length of C S ϱ is defined by AL ST ) : lim n j 1 n ( ) h T zj z j 1, 2.3 if the limit exists for and is indeendent of all described artitions of C S ϱ with F Z 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 suremum or the infimum, resectively.

ISRN Geometry 5 Notice that because h T is in general not a symmetric function, the orientation in the artition may have essential influence onto the value of AL S,T ϱ and is, therefore, assumed here always to be ositive. For studying AL S,T ϱ, let a arameter reresentation of the unit-s-circle C S 1 be given by C S 1 {R S ϕ cos ϕ, sin ϕ T, ϕ<2π}. Later on, we will assume that R S is a.e. differentiable. From the relation h S ( (x, y ) T ) 1, ( x, y ) T CS 1, 2.4 it follows that h S ( (cos ϕ, sin ϕ ) T ) 1 R S ), ϕ<2π. 2.5 In other words, with the notation M S ϕ h S cos ϕ, sin ϕ T,wehave ( ) T cos ϕ sin ϕ C S 1 ( ), ( ), ϕ<2π M S ϕ M S ϕ. 2.6 This motivates the following definition which generalizes more articular notions from earlier considerations. Definition 2.2. For an arbitrary star body S,theS-generalized sine and cosine functions are sin S ) sin ϕ ( ) cos ϕ ( ), cos S ϕ ( ), ϕ, 2π, 2.7 M S ϕ M S ϕ resectively. Notice that there is an elementary geometric interretation of these generalized trigonometric functions when one considers a right-angled triangl Tr, T, x, T, x, y T with x > andy > as follows. The S-generalized sine and cosine of the angle ϕ, 2π between the directions of the ositive x-axes and the line through the oints, T and x, y T are sin S ) y h S ( (x, y ) T ), cos S ) x h S ( (x, y ) T ), 2.8 resectively. These functions satisfy h S ( coss ), sins )) 1, 2.9 generalizing the well-known formula cos 2 ϕ sin 2 ϕ 1.

6 ISRN Geometry Definition 2.3. The S-generalized olar coordinate transformation Pol S :,, 2π R 2 2.1 is defined by x r cos S ), y r sins ), ϕ<2π, r<. 2.11 Let us denote the quadrants in R 2 in the usual anticlockwise ordering by Q 1 u to Q 4. Theorem 2.4. The ma Pol S is almost one-to-one, for x /, its inverse Pol 1 S r h S ( x, y ), is given by ( y ) arctan ϕinq 1,π ϕinq 2,ϕ πinq 3, 2π ϕinq 4, x 2.12 and its Jacobian satisfies J ( r, ϕ ) D( x, y ) D ( r, ϕ ) M 2 S r ( ). 2.13 ϕ Proof. The roof 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 reresentations sin ( ) 1 S ϕ ( ) [ ( ) )] cos ϕm S ϕ sin ϕm ϕ S, M 2 S cos ( ) 1 S ϕ ( ) [ 2.14 ( ) ( )] sin ϕm S ϕ cos ϕm ϕ S ϕ. M 2 S Using S-generalized olar coordinates, we can write C S ) { (ϱ coss ),ϱsins )) T, ϕ<2π, ϱ > }. 2.15 We assume from now on that h T is ositively homogeneous, ut h T ( zj z j 1 ) ht ( (Δj x, Δ j y ) T ), 2.16 and consider ( ( ) Δj x ) h T zj z j 1 ht Δ j ϕ, Δ ( )) T jy ϕ Δj ϕ, Δ j ϕ 2.17

ISRN Geometry 7 for sufficiently thin artition Z n and Δ j ϕ>. We get in the limit, which was assumed in Definition 2.1 to be uniquely determined AL S,T ) 2π ( h T x ),y )) 2π dϕ ϱ ( x ) h T ϱ, y ( )) ϕ dϕ ϱ 2π ( (cos ( ) ϱ h T S ϕ, sin ( )) T ) S ϕ dϕ ϱal S,T 1. 2.18 It follows from the roof of Theorem 2.4 that AL S,T ) ϱ 2π R 2 S ( ) ( ) ϕ ht (M ( ) sin ϕ S ϕ M ( ) ( )) cos ϕ S ϕ dϕ, cos ϕ sin ϕ 2.19 that is, ( ) 2π AL S,T ϱ ϱ R 2 ) ( ( ) ( )) S ht O ϕ xs ϕ dϕ, 2.2 with O ) ( ) R ( ) S ϕ cos ϕ sin ϕ ( ), x S ϕ R 2 ( ) S ϕ sin ϕ cos ϕ 1. ( ) R S ϕ 2.21 The following lemmas and corollaries reresent certain stes towards a reformulation of formula 2.2. Lemma 2.5. In the case of their existence, the artial derivatives h S,x and h S,y x, y h S x, y satisfy the reresentation ( )( ( ( ) ( ))) 1 hs,x r coss ϕ,rsins ϕ ( ( ) ( )) ( O ϕ ) ( ) x S ϕ. 1 h S,y r coss ϕ,rsins ϕ of the function 2.22 Proof. It follows from the relation h S ( r coss ),rsins )) r 2.23 that the artial derivatives h S,x and h S,y satisfy the equation system ϕ h ( ( ) ( )) S coss ϕ, sins ϕ, r h ( ( ) ( )) S r coss ϕ,rsins ϕ 1. 2.24

8 ISRN Geometry Solving this differential equation system, we get ( ( ) ( )) 1 h S,x r coss ϕ,rsins ϕ R 2 ( ) ( ( ) ( ) )) ( ) R S ϕ cos ϕ R S ϕ S sin ϕ, ( ( ) ( )) 1 h S,y r coss ϕ,rsins ϕ R 2 ( ) ( 2.25 ( ) ( ) ( )) ( ) R S ϕ sin ϕ R S ϕ S ϕ cos ϕ. Hence, ( ( ( ) ( ))) ( ) hs,x r coss ϕ,rsins ϕ 1 ( ( ) ( )) ( O ϕ ) ( ) x S ϕ. h S,y r coss ϕ,rsins ϕ 1 2.26 Let B be a 2 2-matrix and BT {B x, y T : x, y T T}. Clearly, multilying the set T by the matrix ( ) 1 1 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 ositively homogeneous h T,differentiable h S,formula 2.2 may be rewritten as AL S,T ) ϱ 2π R 2 ) ( ) ) S h ( ) 1 T( hs x, y x,y PolS r,ϕ dϕ. 2.27 1 Proof. Based uon Lemma 2.5,formula 2.2 may be reformulated as AL S,T ) ϱ 2π R 2 S (( )( ( ( ) ( )) )) ( ) 1 hs,x r coss ϕ,rsins ϕ ϕ ht ( ( ) ( )) dϕ. 1 h S,y r coss ϕ,rsins ϕ 2.28 Because of h T (( 1 1 )( )) { ( )( ) } ξ 1 ξ inf λ>: λt η 1 η { ( ) ( ) } ξ 1 inf λ>: λ T η 1 (( )) ξ h ( ) 1, T 1 η 2.29 it follows the assertion. Remark 2.7. The lug-in version h S x, y x,y PolS 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 olar coordinates, Pol S h S x, y r, ϕ.

ISRN Geometry 9 Proof. Changing Cartesian coordinates x, y with S-generalized olar coordinates r, ϕ, we have r h S x, y and ϕ arctan y/x. Startingfrom x h ( ) S x, y r h ( ) S x, y x r ϕ h ( ) S x, y x ϕ, 2.3 and the analogous one for / y h S x, y, and taking into account that r h ( ) S x, y x,y PolS r,ϕ r h ( ( ) ( )) S r coss ϕ,rsins ϕ 1, ϕ h ( ) S x, y x,y PolS r,ϕ 2.31 ϕ h ( ( ) ( )) S r coss ϕ,rsins ϕ, it follows x h ( ) r S x, y x ( ( ) ( )) x,y Pol S r,ϕ h S,x r coss ϕ,rsins ϕ, 2.32 y h ( ) r S x, y y ( ( ) ( )) x,y Pol S r,ϕ h S,y r coss ϕ,rsins ϕ. Remark 2.8. For ositively homogeneous h T and differentiable Minkowski functional h S of the star disc S, formula 2.2 may be rewritten as AL S,T ) ϱ 2π R 2 ) ( ( ( ))( )) S h ( ) 1 PolS hs T x, y r, ϕ dϕ. 2.33 1 Definition 2.9. AstarbodyS and a star disc T satisfy the rotated gradient condition if ( ) h ( ) 1 T( hs x, y x,y PolS r,ϕ ) 1, a.e. 2.34 1 Letusnoticethatattheoint x, y from C S ϱ, the gradient h S x, y is normal to the level set C S ϱ, ϱ>ofh S. The following lemma is a consequence of the above consideration. Lemma 2.1. For a star body S and a star disc S satisfying the rotated gradient condition 2.34, the ositive directed S -arc-length of C S ϱ allows the reresentation ( ) 2π AL S,S ϱ ϱ R 2 ) s dϕ. 2.35

1 ISRN Geometry We consider now the area function A S ) ϱ 2 A S 1 ϱ2 2 2π R 2 S ) dϕ, 2.36 where A S 1 denotes the area content of the unit disc K S 1. The derivative of the area function satisfies obviously d dϱ A ( ) 2π S ϱ ϱ R 2 ) S dϕ. 2.37 The following theorem has thus been roved. Theorem 2.11. If the star body S and the star disc S satisfy the rotated gradient condition 2.34, then U S ) ALS,S ), 2.38 that is, the S-generalized circumference of S coincides with the ositive directed S -circumference of S. If relation 2.38 holds, then AL S,S 1 2A S 1. 2.39 Consequently, the ratios A S ϱ /ϱ 2 and AL S,S ϱ /2ϱ satisfy the relations ( ) A S ϱ a A S 1 c AL ( ) S,S ϱ, ϱ >. 2.4 2ϱ ϱ 2 In this sense, the geometry and the arc-length measure generated by S fulfill our exectations. 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 ositively homogeneous, h S is a.e. differentiable, and the rotated gradient condition 2.34 is satisfied. Definition 2.12. a The roerties of the star bodies K S ϱ, ϱ>, which are exressed by the equations a and c in 2.4 are called the area content and the S-generalized circumference roerties of the discs, resectively. b The quantity A S 1 : π S is called the S-generalized circle number of the star bodies K S ϱ, ϱ>. We may write now 2.4 as AL S,S ) 2π S ϱ, AS ) π S ϱ 2. 2.41

ISRN Geometry 11 Notice that the circle number function S π S assigns a generalized circle number to any star body K S ϱ satisfying assumtion 2.34. The restrictions of this function to l 2, -balls or axes aligned ellises were considered in Richter 1 3. Examle 2.13. Here, we consider a first, rather general case, where the rotated gradient condition 2.34 is satisfied. Let and d denote a rimary C 1 -norm in R n and the corresonding dual one, resectively. It is roved in Yang 1 that x d 1, x R n. 2.42 Hence, if S is a convex body, that is, h S x x is a rimary norm, and if ( ) 1 T { x R n : x d 1 } S 1 2.43 is the unit ball with resect to the corresonding 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 examle, to Pisier 11, where volumes of convex bodies are dealt with. Alternatively, one may use S-generalized olar coordinates for making the resective calculations in given cases. The articular results for l 2, -circles with 1 and of axes aligned ellises as well as the corresonding generalized circle numbers have been dealt with in Richter 1, 3. Examle 2.14. We consider now the nonconvex l 2, -circles C { x, y R 2 : x y 1} with <<1. Such generalized circles corresond to antinorms. A suitable arc-length measure for measuring C is based uon the star disc S { x, y R 2 : x y 1} with < satisfying 1/ 1/ 1. The star discs S are closely related to secific semiantinorms with resect to the canonical fan. The corresonding generalized circle numbers have been determined in Richter 2. Asbecausethiswasdonewithoutreferring exlicitely to 2.34,wemaystateherethefollowingroblem. Problem 1. Give a general descrition of sets T satisfying condition 2.34 for sets S being generated by antinorms. As was indicated in Richter 2, -generalized circle numbers for <<1 may occur, for examle, within certain combinatorial formulae. Notice further that the recirocal values of the coefficients of the binomial series exansion 1 4 1 4u 1 π 1/n un, n u (, 1 ), 2.44 4 are just the generalized circle numbers corresonding to the nonconvex l 2,1/n -circles. One could also ask for a ossibly elementary geometric? exlanation of this fact.

12 ISRN Geometry 3. Unbounded Star Discs In this section, we consider a class of truncated unbounded Orlicz discs. More generally than in the receding section, a star-shaed 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. Secial sets of this tye will be studied in this section. To be more secific, let us consider, for arbitrary <, the function ( x, y ) ( x, y ) ( x y ) 1/, x /, y/, 3.1 which denotes a semi-antinorm, and the -generalized circle { (x, ) C y 2 R : ( x, y ) } 1. 3.2 The airs of straight lines y 1and x 1 reresent asymtotes for the circle C as x or y, resectively. The intersection oint of the -circle C with the line y x is for ositive coordinates x,y 2 1/ 1, 1. Let further C r r C,r>denotethe-generalized circle of -generalized radius r>. It is the boundary of the unbounded -generalized disc of -generalized radius r, { (x, ) K r y R 2 : x, y } r rk, K K 1. 3.3 As because x, y r x y r f x f ( y ) f r for f λ λ, λ, 3.4 one may call K r a two-dimensional Orlicz antiball corresonding to the Young function f. ThediscK is a star-shaed but noncomact set and, therefore, not a star body. For any x, y C r one may think of r as the value of the Minkowski functional with resect to the reference set K. 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 { (x, ) C x 1 : y 2 R : ( x, y ) ( ) Π 1 x, y ( ) < ( ) ( ) } x1,y 1 Π1 x1,y 1 ( ) Π 1 x, y Π 1 x1,y 1 { } (x, ) x y y 2 R : x y < x1 y 1, x 1 y 1 3.5 where denotes Euclidean norm, 1 1, 1, x 1 is chosen according to x 1 >x 2 1/ and y 1 1 x 1 1/ < 1.

ISRN Geometry 13 The question of interest is now whether we may define in a reasonable way circle numbers for the truncated -discs K x 1 r rk x 1,theboundariesofwhicharethe-circles C x 1 r rc x 1 of -radius r, where K x 1 : K C x 1, C x 1 : C C x 1. 3.6 To this end, let Z n z,z 1,...,z n be an arbitrary successive anticlockwise-oriented artition of the truncated circle C x 1 satisfying z x, 1 x 1/ and z n x 1, 1 x 1 1/.Weconsider the sum S Z n n zj z q j 1 j 1 n ( xj x j 1 q yj y j 1 q) 1/q, q, 1, 3.7 j 1 and observe that due to the reverse triangle inequality it decreases monotonously as F Z n : su z j z q j 1. 1 j n 3.8 AccordingtothesymmetryofC x 1, the following remark is justified. Remark 3.1. For q, 1,thel 2,q -arc-length of the truncated circle C x 1 is defined as AL x 1,q 8 lim F Z S Z n. n 3.9 then If x y x denotes an arbitrary arameter reresentation of the truncated circle C x 1, 1 8 ALx 1,q lim F Z n j 1 ( n Δy j q) 1/q x1 ( 1 Δx Δx j j 1 y x q ) 1/q dx. 3.1 x Let us denote the usual Euclidean area content of the truncated circle disc K x 1 by A x 1. Lemma 3.2. Let for arbitrary < the number 1/ 1/ 1.Then,, 1 be uniquely defined by the equation A x 1 1 2 ALx 1,. 3.11 Proof. With y x ( 1 x ) 1/ 1 x 1/, y x 1 x 1/ 1 x 1, 3.12 it follows from the above formulae that x1 ( AL x 1,q 8 1 1 x 1/ 1 q x 1 q) 1/q dx. 3.13 1/2 1/

14 ISRN Geometry Changing variables u x,dx du/ u 1 / causes a change of the limits of integration: AL x 1,q 8 1/2 x 1 (1 1 u 1 / q u 1 / q) 1/q du 8 1/2 ( u 1 / q 1 u 1 / q) 1/q du. x 1 u 1 / 3.14 Assuming now 1/ 1/q 1, or equivalently q / 1 :, it follows that AL x 1, 8 1/2 x 1 (u 1 1 u 1) 1 / du 8 1/2 x 1 ( 1 u u u 1 u )1 1/ du. 3.15 Hence, AL x 1, 8 1/2 x 1 u 1/ 1 1 u 1/ 1 du. 3.16 Now, what about the area content of the truncated circle C x 1?Thel 2, -generalized standard triangle coordinate transformation Tr from Richter 4 is defined by Tr ( r, μ ) ( x, y ) with x rμ, y r ( 1 μ ) 1/. 3.17 Because of {( x, y ) : x y 1,x x x 1 } Tr {1} x,x 1, r {( x, y ) : x y 1, x x x 1 } {( rx,ry ) : x y } 1, x x x 1 { (ξ, ) ξ η : r η r 1, x ξ } r x 1 { (x, ) y : x y r,x x } ( ) : μ x1 r 3.18 Tr {r} x,x 1, it follows r {( x, y ) : x y } 1, x x x 1 Tr, 1 x,x 1 K x 1 r 1. 3.19 That is, K x 1 Tr, 1 x,x 1. 3.2

ISRN Geometry 15 Changing Cartesian with standard triangle coordinates in the integral A x 1 K x 1 d ( x, y ), 3.21 we get A x 1 8 1 r ( x1 r ( ) 1 μ ) 1 / dμ dr 8 μ x 2 x1 2 1/ ( 1 μ ) 1/ 1 dμ. 3.22 Substituting y μ,dy/dμ y 1 /, it follows that A x 1 4 x 1 1/2 ( ) 1/ 1y 1 y 1/ 1 dy 4 1/2 x 1 y 1/ 1( 1 y ) 1/ 1 dy. 3.23 Hence, the lemma is roved. Remark 3.3. Because of <, A x 1 as x 1. 3.24 The following corollary and definition are now quite obvious and well motivated. Corollary 3.4. For arbitrary x 1 >x 2 1/, the truncated star discs K x 1 -generalized circumference roerties a and c, resectively, from which it follows immediately A x 1 r a A x c 1 r 2 ALx 1 2r, r have the area content and, 3.25 d dr Ax 1 r AL x 1, r. 3.26 Remark 3.5. One may think of 3.26 as reflecting a generalized method of indivisibles for each x 1 > x, where the truncated circles C x 1 are the indivisibles and measuring them is based uon the geometry generated by the disc K. Definition 3.6. For arbitrary x 1 > 2 1/, the quantity A x 1 number of the truncated circle C x 1. : π x 1 will be called the circle References 1 W.-D. Richter, On l 2, -circle numbers, Lithuanian Mathematical Journal, vol. 48, no. 2,. 228 234, 28.

16 ISRN Geometry 2 W.-D. Richter, On the π-function for nonconvex l 2, -circle discs, Lithuanian Mathematical Journal,vol. 48, no. 3,. 332 338, 28. 3 W.-D. Richter, Ellises numbers and geometric measure reresentations, to aear in Journal of Alied Analysis. 4 W.-D. Richter, Generalized sherical and simlicial coordinates, Journal of Mathematical Analysis and Alications, vol. 336, no. 2,. 1187 122, 27. 5 M. Moszyńska and W.-D. Richter, Reverse triangle inequality. Norms and anti-norms, submitted. 6 D. Hilbert, Mathematische Probleme, Mathematikerkongress, Paris, France, 19. 7 R. Webster, Convexity, The Clarendon Press, Oxford University Press, New York, NY, USA, 1994. 8 M. Moszyńska, Selected Toics in Convex Geometry, Birkhäuser, Boston, Mass, USA, 26. 9 A. C. Thomson, Minkowski Geometry, vol.63ofencycloedia of Mathematics and Its Alications, Cambridge University Press, Cambridge, UK, 1996. 1 W. H. Yang, On generalized Hölder inequality, Nonlinear Analysis: Theory, Methods & Alications, vol. 16, no. 5,. 489 498, 1991. 11 G. Pisier, The Volume of Convex Bodies and Banach Sace Geometry, vol.94ofcambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1989.

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