On the approximation of a convex body by its radial mean bodies

Transcription

1 Available olie at wwwtjsacom J Noliea Sci Al 9 6, Reseach Aticle O the aoximatio of a covex body by its adial mea bodies Lvzhou Zheg School of Mathematics ad Statistics, Hubei Nomal Uivesity, 435 Huagshi, P R Chia Commuicated by R Saadati Abstact I this ae, we coside the aoximatio oblem o the volume of a covex body K i R by those of its adial mea bodies R K Secifically, we establish the idetity log R K = whe K is a ellisoid i R c 6 All ights eseved +, Keywods: Covex body, adial mea body, diffeece body, esticted chod ojectio fuctio MSC: 53A4, 4A5 Itoductio I covex geomety, coesodig to each covex body K R, thee ae two imotat geometic objects called diffeece body DK ad ola ojectio body Π K The diffeece body was studied by Mikowski, ad has foud may alicatios i mathematical hysics ad PDEs See, fo examle, the books of Badle [] ad Kawohl [] Pojectio bodies also oigiated i the wok of Mikowski, ad ae widely used i the local theoy of Baach saces, stochastic geomety, mathematical ecoomics, ad othe aeas [4, 9] The ola ojectio body, the ola body of the ojectio body, aeas exlicitly i the moe ecet liteatue; its behavio ude liea tasfomatios ofte edes it moe atual tha the ojectio body itself Both the diffeece body ad the ola ojectio body aea i kow affie iequalities The fist is a igediet i the famous Roges - Shehad iequality [6, ], that is, V DK V K addess: oasiszlz@siacom Lvzhou Zheg Received 5--6

2 L Zheg, J Noliea Sci Al 9 6, The secod aeas i the celebated Zhag ojectio iequality [8, 7, 8] V K V Π K, whee the equality holds i as well as i, if ad oly if K is a simlex Give a covex body K R ad >, Gade ad Zhag [8] oigially itoduced a imotat geometic body, called adial -th mea body R K of K, whose adial fuctio is defied by ρ RKu = ρ K x, u dx, u S 3 V K K It is emakable that the bodies R K fom a sectum likig the diffeece body DK of K ad the ola ojectio body Π K of K, which coesod to = ad =, esectively Moe imotatly, fo < < q, the followig stog ad sha affie iequality V DK c,qv R q K c,v R K V K V Π K, 4 which was established i [8], imlies the above metioed Roges-Shehad iequality ad Zhag ojectio iequality as secial cases I 4, each equality holds if ad oly if K is a simlex, ad c, = B +, is a costat Secifically, whe = ad q, the middle iequality i 4 becomes the Roges-Shehad iequality, ad whe ad q =, it becomes the Zhag ojectio iequality Theefoe, i some sese, adial mea bodies R K exhibit a stog uity i covex geomety I [6], the authos established the idetity elated chod owe itegals of covex body K ad dual quemassitegals of R K It is oved i [8] that fo, the adial -th mea body R K is a oigi-symmetic covex body Now, a oblem is atually asked, Poblem If K is a covex body i R, how about the ate of aoximatio o the volume V K of covex body K by the volume V R K of its adial -th mea body R K? It is oted that the aoximatio oblem of a covex body by its associated bodies, such as floatig bodies, covolutio bodies, ad cetoid bodies, ojectio bodies etc, have bee itesively ivestigated We efe to eg [5, 8,,, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5] fo futhe details, extesios ad alicatios As a aside, we obseve that thoughout the whole ae [8], all affie iequalities attai extemum if ad oly if the covex body is a simlex Theefoe, it will atually lead us to study the adial mea bodies R K whe K is a oigi-symmetic covex body Fom ow o, we shall use to eeset the dimesioal volume V of a covex body i R I this ae, fo the affie ivaiat atio RK, we will ove the followig theoems Theoem Suose K is a ellisoid i R ad R K is the adial -th mea body of K The log R K = + 5 Theoem Suose K is a simlex i R The log R K = 6 This ae is ogaized as follows I Sectio, we develo some otatio ad list, some basic facts egadig covex bodies Good geeal efeeces fo the theoy of covex bodies ae ovided by the books of Gade [7] ad Scheide [] I Sectio 3, we give some bouds fo the aoximatio of volume i the case of a geeal covex body The oofs of Theoems ad will be aaged i the Sectio 4

3 L Zheg, J Noliea Sci Al 9 6, Notatios ad Peiaies The settig fo this ae is dimesioal Euclidea sace, R A covex body K i R is a comact covex set that has a o-emty iteio As usual, S deotes the uit shee, B the uit ball ad o the oigi i R The volume of B is deoted by ω If u S, we deote by u the -dimesioal subsace of R othogoal to u ad by l u the lie though o aallel to u We wite V k fo the k dimesioal Lebesgue measue i R Let K be a covex body i R The adial fuctio ρ K x, of K with esect to x R, is defied by ρ K x, u = max{c : x + cu K}, u S If x is the oigi, we usually deote ρ K o, u by ρ K u Fo u S ad y u, let X u Ky = V K l u + y, the fuctio is called the X ay of K i the diectio u See [7] fo details Let E K, u = {y u : X u Ky } ad a K, u = V E K, u fo, ad u S I [7] the fuctio a K, u is called the esticted chod ojectio fuctio of K Note that if u S, the E K, u = K u ad a K, u = V K u, ad whe > ρ DK u, we have E K, u = ad a K, u = The diffeece body of the covex body K, deoted by DK, is the cetally symmetic covex body ceteed at the oigi defied by It is ot difficult to veify that DK = K + K = {x y : x, y K} ρ DK u = max x K ρ Kx, u = max y u V K l u + y, u S A ofte used fact i both covex ad Baach sace geomety is that associated with each covex body K i R is a uique ellisoid JK of maximal volume cotaied i K The ellisoid is called the Joh ellisoid of K ad the cete of this ellisoid is called the Joh oit of K The Joh ellisoid is extemely useful; see, fo examle, [, 7]ad [] fo alicatios Two imotat esults coceig the Joh ellisoid ae Joh s iclusio ad Ball s volume-atio iequality Joh s iclusio states that if K is a oigi-symmetic covex body i R, the JK K JK Amog a slew of alicatios, Joh s iclusio gives the best ue boud,, fo the Baach-Mazu distace of a dimesioal omed sace to dimesioal Euclidea sace Ball s volume atio iequality is the followig: if K is a oigi-symmetic covex body i R, the ω, with equality if ad oly if K is a aallelotoe The fact that thee is equality i oly fo aallelotoes was established by Bathe [3] Lemma Makov s Iequality Suose X,, µ is a measue sace, f is a measuable exteded eal-valued fuctio ad ε > The µ{x X : fx ε} f dµ 3 ε The ivaiat oety of adial -th mea body ude o-sigula liea tasfomatio shows that they ae atual objects i affie geomety X

4 L Zheg, J Noliea Sci Al 9 6, Lemma [6] Let K be a covex body i R ad GL the osigula liea tasfomatio gou The fo ϕ GL ad >, we have R ϕk = ϕr K Lemma 3 [8] Let K be a covex body i R ad let u S The fo >, we have We will also use the followig lemma Lemma 4 [8] Let > The K ρ K x, u dx = ρdk u a K, u d 4 B +, = log + 4 logγ + log 3 logγ log ± o 5 3 Geeal Bouds Lemma 3 Let K be a covex body i R ad u S The whee the equality holds i the left had if ad oly if a had if ad oly if = Poof The ight iequality is obvious Sice a K, u ρ DK u V K u a K, u V K u, 3 K a K, u = ak ρ DK u ρ DKu +, u is liea i, the equality holds i the ight is cocave i, we have ρ DK u ρ DK u a Kρ DK u, u + a K, u ρ DK u = V K u ρ DK u, u ρ DK u a K, u The equality coditio ca be deived fom the agumets easily This comletes the oof Lemma 3 Let K be a covex body i R ad u S Fo < <, we have Fo >, we have + B +, ρ RKu ρ DK u B +, ρ R Ku ρ DK u + Poof Accodig to the fomula = ρ DK u a K, ud ad 3, we have = ρdk u a K, ud ρ DK ua K, u = ρ DK uv K u,

5 L Zheg, J Noliea Sci Al 9 6, ad = ρdk u ρdk u a K, ud = ρ DKuV K u V K u d ρ DK u It yields that ρ DKuV K u ρ DK uv K u 3 Combied with 3 ad 3, we have O the othe had, we have This comletes the oof ρ R K u = ρ R K u = = ρdk u ρdk u ρ K x, u dx K ρdk u a K, ud V K u + ρ+ DK u + ρ DK u a K, ud = V K u ρ + DK u ρ DK ub +, Fom Lemma 3, we ca get immediately that, V K u d ρ DK u s s ds Coollay 33 Let K be a oigi-symmetic covex body i R ad u S Fo < <, we have Fo >, we have + B +, ρ R Ku ρ DK u B +, ρ R Ku ρ DK u + By usig Makov s iequality, we ca obtai a ew ue boud fo ρ RKu ρ DK u Lemma 34 Let K be a covex body i R ad u S If >, we have ρ RKu ρ DK u 33

6 L Zheg, J Noliea Sci Al 9 6, Poof Accodig to Makov s iequality 3, we have With Lemma 3, it yields ρ R K u = This comletes the oof a K, u = V {y K u : X u Ky } X u Kydy = K u K ρ K x, u dx ρdk u d = ρ DK u Theoem 35 Let K be a covex body i R ad > The DK log DK R K DK 34 Poof We have that ad Fom Lemma 34, it follows DK R K = ρ DKu ρ R Ku du S = ρ DKu ρ R u K S ρ DK u du ρ R K u ρ DK u Fom Lemma 3 ad Lemma 4, it imlies ρ R u K ρ DK u B +, = log By usig Lebesgue covegece Theoem, it gives log DK R K = = log + ± o log 4 logγ + log 3 logγ log ± o = DK, log DK R K = = DK, S ρ DKu S ρ DKu S ρ DKu S ρ DKu which yields the equied iequalities This comletes the oof ρ R u K log ρ DK u log log du ± o log ρ R u K log ρ DK u log log du ± o log du du

7 L Zheg, J Noliea Sci Al 9 6, If covex body K is oigi-symmetic, the DK = K + K = K Hece, we have: Coollay 36 Let K be a oigi-symmetic covex body i R ad > The 4 Poof of Mai Results log R K Fistly, we ove Theoem, which is ivolved i a lage umbe of estimatios Poof of Theoem We fist coside the adial -th mea body of the uit ball B Obviously, we have a B, u = V fo all u S Fom Lemma 3, it gives fo all > Let we have ρ R B u = B, ρ DB u =, ρ B x, u dx V B B = V B = ω ω = ω ω ρdb u a B, u d 4 d d = ω B +, + ω d, = { ω B +, + ω }, ρ RB u = d,, which is a costat deedig oly o the umbes ad Hece the adial -th mea body of uit ball B is still a ball ceteed at the oigi with adius d,, ie, R B = d, B Whe K is a ceteed ellisoid, thee exists ϕ GL such that K = ϕb Fom Lemma ad the above agumet, we have which is still a ceteed ellisoid Now, we have R K = R ϕb = ϕr B = ϕd, B = d, ϕb = d, K, log R K = = log R B B [ ω log B +, + ω ] With the fact that, Γx = [ πx x e x + x + ] 88x ± ox, x,

8 L Zheg, J Noliea Sci Al 9 6, we have Sice ω B +, + ω log R K = = ω log [ Γ + π + e + π π e ++ Γ + = π ad each tem ca be comuted as + e + ++ B +, + ω + + [ ] ± o + ± o, ] ± o Γ + π + e = e log Γ+ π = + + log Γ + log Γ π + + ± o, π + ++ = = ± o, So we have ω ++ + B +, + ω = e + log ++ + = log log ± o = + log Γ + + π log log log + + log Γ π [ + log Γ ] ± o π Cosequetly, it yields [ ω log B +, + ω ] = +

9 L Zheg, J Noliea Sci Al 9 6, This comletes the oof Poof of Theoem By usig the left iequality i 4 ad its equality coditio, it has DK = c, R K, hee Moeove, c, = B +, log DK R K = = = log DK c, log DK B +, log DK log log = log DK = DK Whe K is a simlex, the equality holds i Roges-Shehad iequality That is, DK = So the desied idetity is obtaied This comletes the oof Fially, we discuss the case whe K is a geeal oigi-symmetic covex body i R The followig lemma ca be deived though the defiitio of ρ RK diectly Lemma 4 Let K ad K be covex bodies i R If K K ad >, the ρ RK u K K ρ RK u, fo all u S By usig the imotat Joh s iclusio ad Theoem, we have JK ρ RKu ρr JKu ρrjku ρrjku, u S, which imlies So it gives R K R K R JK R JK = R JK

10 L Zheg, J Noliea Sci Al 9 6, Hece, it has R K Cosequetly, fom Lemma 4 it yields R JK log R K log R JK = + 4 Similaly, by usig Joh s iclusio, followed by Lemma 4 ad Ball s volume-atio iequality, it gives which imlies So it gives Hece, it has R K ρ RJKu ω R JK ρrku ρrku, u S, ω R K ω R JK = ω R JK JK ω R K Cosequetly, fom Lemma 4 it yields R JK log ω R K log R JK = + 4 Combied with 4 ad 4, it yields log ω R K + log R K Ackowledgemets The eseach is atially suoted by Hubei Povicial Deatmet of Educatio No 336 Refeeces [] K Ball, Volume atios ad a evese isoeimetic iequality, J Lodo Math Soc, 44 99, [] C Badle, Isoeimetic iequalities ad alicatios, Pitma, Lodo, 98 [3] F Bathe, O a evese fom of the Bascam-Lieb iequality, Ivet Math, , [4] J Bougai, J Lidestauss, Pojectio bodies, Geometic Asects of Fuctioal Aalysis, Lectue Notes i Math, , 5 7 [5] S Cami, P Gochi, The L Busema-Petty cetoid iequality, Adv Math, 67, 8 4 [6] G Chakeia, Iequalities fo the diffeece body of a covex body, Poc Ame Math Soc, 8 967, [7] R Gade, Geometic Tomogahy, secod editio, Cambidge Uivesity Pess, Cambidge, 6,

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