Dec Communication on Applied Mathematics and Computation Vol.32 No.4

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1 » Å 32 Å 4 Dec Commuicatio o Applied Mathematics ad Computatio Vol.32 No.4 DOI /j.iss L p Blaschke additio for polytopes LAI Ducam 1,2 (1. College of Scieces, Shaghai Uiversity, Shaghai , Chia; 2. Departmet of Mathematics, Yebai Teacher s Traiig College, Yebai , Vietam) Abstract Usig the solutio of the geeral L p Mikowski problem for polytopes, the L p Blaschke additio for polytopes (ot origi symmetric) is itroduced ad the Lutwak s L p Blaschke additio for the origi symmetric covex bodies is exteded. I additio, the correspodig L p Keser-Süss iequality for the polytopes is established. Key words Blaschke additio; L p Blaschke additio; L p Mikowski problem; L p Keser-Süss iequality 2010 Mathematics Subject Classificatio 52A20; 52A40 Chiese Library Classificatio O186.5 ½ L p Blaschke ÂÁÃ 1,2 (1. ¹ ; 2. ¹ ² , º ) µ L p Blaschke Lutwak Æ ¼ «L p Blaschke ³ L p Keser-Süss Blaschke L p Blaschke L p Mikowski ± L p Keser-Süss A20; 52A40 À O186.5 A ¾ (2018) Itroductio The operatio betwee covex bodies ow called the Blaschke additio goes back to Mikowski [1], at least whe the bodies are polytopes. Later, Blaschke [2] foud a defiitio Received ; Revised Supported by the Natioal Natural Sciece Foudatio of Chia ( ); the Shaghai Leadig Academic Disciplie Project (J50101) Correspodig author LAI Ducam, research iterests are covex geometry ad geometry aalysis. am.laiduc@gmail.com

2 950 Commuicatio o Applied Mathematics ad Computatio Vol. 32 suitable for the smooth covex bodies i R 3. The moder defiitio, appropriate for ay pair of covex bodies, was proposed by Fechel ad Jesse [3] by combiig the developmet of surface area measures. The Blaschke additio has foud may applicatios i geometry [4-15]. The L p Blaschke additio for ay pair of origi symmetric covex bodies was defied by Lutwak [12], by usig the solutio of the eve L p Mikowski problem. I this paper, we exted the L p Blaschke additio for ay pair of covex polytopes cotaiig the origi i their iterior. From this defiitio, we exted Lutwak s L p Keser-Süss iequality ad show the expoet is best (see Theorems 2 ad 3). Furthermore, a applicatio of L p Keser-Süss iequality is preseted. This paper is orgaized as follows. I Sectio 1, we collect some basic otatios ad itroduce the L p Blaschke additio for polytopes. I Sectio 2, we establish the L p Keser- Süss iequality. 1 Notatios ad prelimiaries For geeral referece, the reader may wish to cosult the books of Garder [16] ad Scheider [17]. Let K deote the space of compact covex subsets of R with oempty iteriors, ad P deote the subset of covex polytopes. The members of K are called covex bodies. We write K o for the set of covex bodies which cotai the origi as a iterior poit, ad put P o = P K o. For a covex body K, let h K = h(k, ) : R R deote the support fuctio of K; i.e., for x R, let h K (x) = max x, y, where x, y is the stadard ier product of x ad y K y i R. We shall use V (K) to deote the -dimesioal volume of a covex body K i R. For K K, let F(K, u) deote the support set of K with the exterior uit ormal vector u, i.e., F(K, u) = x K : x, u = h(k, u). The ( 1)-dimesioal support sets of a polytope P P are called the facets of P. If P P has facets F(P, u i ) with areas a i, i = 1, 2,, m, the S(P, ) is the discrete measure S(P, ) = m a i δ i with the (fiite) support {u 1, u 2,, u } ad S(P, {u i }) = a i, i = 1, 2,, m; here δ i deotes the probability measure with the uit poit mass at u i. i=1 For a Borel set ω S 1, the surface area measure S K (ω) = S(K, ω) of the covex body K is the ( 1)-dimesioal Hausdorff measure of the set of all boudary poits of K for which there exists a ormal vector of K belogig to ω. For p 1, it was show i [12] that correspodig to each covex body K K o, there is a positive Borel measure o S 1, the L p surface area measure S p (K, ) of K, such that

3 No. 4 LAI Ducam: L p Blaschke additio for polytopes 951 for every L Ko, V p (K, L) = 1 h p (L, u)ds p (K, u). (1) S 1 Moreover, V p (K, K) = V (K). Here, the L p surface area measure is absolutely cotiuous with respect to S(K, ), S p (K, ) = h 1 p (K, )S(K, ). The L p Mikowski iequality [12] states: if K K o ad p > 1, the with the equality if ad oly if K ad L are dilates. V p (K, L) V p (K)V p (L) (2) A importat biary operatio i the covex geometry is the Blaschke additio, defied for covex bodies K ad L i R by lettig K L be the uique covex body with cetroid at the origi such that problem. S(K L, ) = S(K, ) + S(L, ). I [18], Hug, et al., established the solutio to the discrete-data case of the L p Mikowski Theorem 1 Let vectors u 1, u 2,, u S 1 that are ot cotaied i a closed hemisphere ad real umbers α 1, α 2, α > 0 be give. The, for ay p > 1 with p, there exists a uique polytope P P o such that m α uj δ uj = h 1 p (P, )S(P, ). j=1 From this theorem, we ca defie the L p Blaschke additio for polytopes: for K, L P o, p > 1, the L p Blaschke additio K p L P o of K ad L is defied by S p (K p L, ) = S p (K, ) + S p (L, ). (3) Note that the L p Blaschke additio for K, L K e is previously defied by Lutwak [12], which relies o his solutio of the eve L p Mikowski problem. 2 L p Keser-Süss iequality I [12], Lutwak obtaied the followig L p Keser-Süss iequality for origi symmetric covex bodies. By the same argumets, we obtai the L p Keser-Süss iequality for the polytopes.

4 952 Commuicatio o Applied Mathematics ad Computatio Vol. 32 Theorem 2 If K, L P o ad 1 < p, the V p (K p L) V p p (K) + V (L) with equality if ad oly if K ad L are dilates. Proof From (1) ad (3), we have Together with (2) yields V p (K p L, Q) = V p (K, Q) + V p (L, Q). V p (K p L, Q) V p p (Q)[V p (K) + V (L)] with equality (for p > 1) if ad oly if K, L, ad Q are dilates. The result follows by takig K p L for Q. For K P o, p > 1, defie pk P e by S p ( p K, ) = 1 2 S p(k, ) S p( K, ). (4) Corollary 1 If K P o ad 1 < p, the with equality if ad oly if K is cetered. V ( p K) V (K) Ispired the excellet survey of Alexadrov, et al. [19] for the Blaschke additio, we ca adapt their argumets for the L p Blaschke additio. The followig elemetary lemma ca be see i [19, Lemma 8.2]. Lemma 1 For all a 1 ad x > 0, the followig iequality holds: (1 + x) a 1 + x a. (5) Moreover, for all 0 < a < 1, there exists x > 0 such that (5) fails. Theorem 3 For every a 1 ad K, L P o iequalities hold: V a ap (1 < p, 2), the followig (K p L) V a ap (K) + V a ap (L). (6) Moreover, for 0 < a < 1, there exist K, L Po such that (6) fails. Proof For a 1, by Theorem 2 ad (5), we have V a( p) (K p L) V a( p) (K) that proves (6) for a 1. (1 + V p (L) V p (K) ) a 1 + V a( p) (L) V a( p) (K),

5 No. 4 LAI Ducam: L p Blaschke additio for polytopes 953 Now, let 0 < a < 1, Z = [ 1, 1] 1, K = rz, L = RZ. The, for i = 1, 2,,, h(k, e i ) = r, h(l, e i ) = R, S(K, e i ) = (2r) 1, S(K, e i ) = (2R) 1. Thus, h(k p L, e i ) 1 p S(K p L, e i ) = h 1 p (K, e i )S(K, e i ) + h 1 p (L, e i )S(L, e i ) = r 1 p (2r) 1 + R 1 p (2R) 1. Note that for v ±e i, i = 1, 2,,, S(K, v) = 0 ad S(L, v) = 0. Thus, K p L = tz for some t > 0. Moreover, h 1 p (K p L, e i )S(K p L, e i ) = h 1 p (tz, e i )S(tZ, e i ) = t p 2 1 = r 1 p (2r) 1 + R 1 p (2R) 1. Cosequetly, we have K p L = tz = (r p + R p ) 1/( p) Z. From (6), we have V a( p) (tz) V a( p) (rz) + V a( p) (RZ), or (r p + R p ) a r a( p) + R a( p), ( ( r ) ( p) ) a ( r ) a( p) R R However, by Lemma 1, for 0 < a < 1, the latter iequality certaily fails for some x = (r/r) p. Fially, we show a applicatio of the L p Keser-Süss iequality for polytopes. Theorem 4 Let K, L P o ( 2) ad let the L p surface area (1 < p ) measure of K do ot exceed the surface area measure of L, that is S p (K, ) S p (L, ). The, { V (K) V (L), 1 < p <, V (K) V (L), p >. Proof Take t (0, 1), ad let µ( ) = S p (L, ) ts p (K, ). Sice t < 1, for every subset ω S 1 such that S p (K, ω) > 0, we have µ(ω) > 0. Sice S p (K, ) is ot cotaied i a closed hemisphere of S 1, we coclude that µ is ot cotaied i a closed hemisphere of S 1. By Theorem 1, there exists M P o whose surface area fuctio coicides with µ, that is µ( ) = S p (M, ). However, S p (L, ) = S p (M, )+ts p (K, ). Therefore, L = M p (t 1/( p) K). By Theorem 2, we the have V p p (L) = V (M p (t 1/( p) K)) V p p 1 (M) + V (t p K) = V 1 p p p (M) + t (V (K)) tv (K).

6 954 Commuicatio o Applied Mathematics ad Computatio Vol. 32 Tedig t to 1, we get V p (L) V p (K), which completes the proof. Remark 1 The L p Blaschke additio (3) i this paper also applies to K K o for p >. I this case, the existece of the L p Blaschke additio is guarateed by the solutio of the geeral L p Mikowski problem [18]. However, for 1 < p <, the origi may lie o the boudary of K by the solutio of the geeral L p Mikowski problem, which will lead to h(k, v) = 0 for some v S 1. Refereces [1] Mikowski H. Allgemeie Lehrsätze über die covexe Polyeder [J]. Nachrichte vo der Gesellschaft der Wisseschafte zu Göttige, 1897: [2] Blaschke W. Kreis ud Kugel [M]. 2d ed. Berli: De Gruyter, [3] Fechel W, Jesse B. Megefuktioe ud kovexe Körper [J]. Daske Vid Selsk Math-Fys Medd, 1938, 16: [4] Campi S, Colesati A, Grochi P. Blaschke-decomposable bodies [J]. Israel J Math, 1998, 105: [5] Firey W J, Grübaum B. Additio ad decompositio of covex polytopes [J]. Israel J Math, 1964, 2: [6] Garder R J, Parapatits L, Schuster F E. A characterizatio of Blaschke additio [J]. Adv Math, 2014, 254: [7] Haberl C. Blaschke valuatios [J]. Amer J Math, 2011, 133: [8] Kiderle M. Blaschke ad Mikowksi edomorphisms of covex bodies [J]. Tras Amer Math Soc, 2006, 358: [9] Klai D A. Coverig shadows with a smaller volume [J]. Adv Math, 2010, 224: [10] Keser H, Süss W. Die Volumia i lieare Schare kovexer Körper [J]. Mat Tidsskr, 1932, 1: [11] Ludwig M. Valuatios o Sobolev spaces [J]. Amer J Math, 2012, 134: [12] Lutwak E. The Bru-Mikowski-Firey theory. I. Mixed volumes ad the Mikowski problem [J]. J Differetial Geom, 1993, 38: [13] Schuster F E. Covolutios ad multiplier trasformatios of covex bodies [J]. Tras Amer Math Soc, 2007, 359: [14] Zouaki H. Represetatio ad geometric computatio usig the exteded Gaussia image [J]. Patter Recogitio Letters, 2003, 24(9/10): [15] Zouaki H. Covex set symmetry measuremet usig Blaschke additio [J]. Patter Recogitio, 2003, 36(3): [16] Garder R J. Geometric Tomography [M]. 2d ed. New York: Cambridge Uiversity Press, [17] Scheider R. Covex Bodies: the Bru-Mikowski Theory [M]. Cambridge: Cambridge Uiversity Press, [18] Hug D, Lutwak E, Yag D, Zhag G. O the L p Mikowski problem for polytopes [J]. Discrete Comput Geom, 2005, 33: [19] Alexadrov V, Kopteva N, Kutateladze S S. Blaschke additio ad covex polyhedra [J]. Tr Semi Vektor Tezor Aal, 2005, 26: 8-30.