207 Ä 9 3 3 Ì Sept 207 Commuicatio o Applied Mathematics ad Computatio Vol3 No3 DOI 03969/iss006-633020703006 ³ Ball-Barthe ƺ ÌÍË (¹ 200444 Á ËÒÉØ˲¾ÝÀÖÜ Ball-Barthe ØÀÉ ¹¾Â¼ Ball-Barthe Ø ÔË²Î¹Æ Â¼ Ball-Barthe ØÀÉÉË Ë²¾ÅÅ Æ Å ÒÉØ Ball-Barthe Ø Â¼ Ball-Barthe Ø 200 ½ 52A40 É O84 Ç A  006-6330(20703-036- Reverse Ball-Barthe iequality SHANG Xiaoma (College of Scieces, Shaghai Uiversity, Shaghai 200444, Chia Abstract By usig mixed discrimiats, a ew proof of the celebrated Ball- Barthe iequality i covex geometry aalysis is preseted The reverse Ball- Barthe iequality ad its equality coditio are established As a applicatio, we estimate the lower boud for the volume of a covex body Key words mixed discrimiat; Ball-Barthe iequality; reverse Ball-Barthe iequality 200 Mathematics Subect Classificatio 52A40 Chiese Library Classificatio O84 0 Ø ÛÜÕ± Ø Ø Û S (D ω V (D, ¼¼ D ÛÈ µ S(D V (D Ù ÆÜ R ÕÒ D Ö Ö ω = π 2 /Γ( + 2 Û R Õß» Ö R ÕÊÖ ÍÒÕ Ö ½ ¼ 204-06-5; 205-0-26 ² ĐÐÕ Ýº (0063 Êà ½ÞÁ E-mail: shagxiaoma@26com
3 Ì ¾ ¼ Ball-Barthe Ø 37 ÜÒà Á»Ø ÅÞ½Þ Â»ÓË Ø Ð Ø Á»Ø Ø Á»Ø Í Û Á»Ø Û̾ Mahler [] 939 à ± Ð K ÛÏ ¼ K K ÛË Ó µ K ÊØ ÕÛ Blaschke-Sataló V (KV (K 4!, ( K = {x R : x y, y K}, x y ÙÆÜ R ÕßÀÖ Mahler ÛÜ V (KV (K ω 2 Á» ÍÕ K ÛÏ ¼¼ K Û Ù Mahler Á»Ø ÛÈ ÍÚ ÏÍ ÍÎñ ½ 990 à ½Ä Ball [2] ³ ÅÜÕ Joh Joh ³ Brascamp-Lieb ½º Đ Barthe [3] ½Á» Brascamp-Lieb ½½ÜÕ³ Ball- Barthe ² 2004 à Lutwak [4] ½Ï»» Ball- Barthe Û (Ball-Barthe Ð t : (0, Û» µ» ( det t(u u u dµ(u ¼¼ u,, u supp µ, Ù µ Ù { } exp log t(u dµ(u, t(u i ÛÈ ÍÕ supp µ Ñ Ê» Ball-Barthe Î Î ½ Æ Á»Ø [4-7] ÔÏ Ball-Barthe Á»Ø Õ Ê ÊÑÈ Ê±½ Ball-Barthe È ½Á» Ball-Barthe ÚÍ 2 (Á» Ball-Barthe Ð t : (0, Û»» ( det t(u u u dµ(u t (u dµ(u, S ¼¼ c > 0, Ö Ã u supp µ, t(u = c µ
38 3 Á» Ball-Barthe ÈÊ 3 K K o, ʱ½ ÖÈ Ïß ( V (K ω det ρ K (u u u ds(u, ¼¼ K ÛÈÕ À ƹ ÆÜ R Õ R Õ Ü K ÖÊ V (K Ù» B = {x R : x } ÖÊ ω Ù ω = π 2 /Γ( + 2 Ð Ù R Õ» Ko Ù R Õ À (ÍÀÒ Ù Ù S( Û Lebesgue Ð K Ko, Í» ρ K(x Ä ρ K (x = max {λ 0 : λx K}, x R \ {0} Ð µ Û ÈÂ͵ Borel [µ] = u u dµ(u, [µ] ÛÈ Å ÚÕ Ã x R, Í x [µ]x = x u 2 dµ(u, (2 µ x u ÙÆÜ R ÕßÀÖ (2 Õ x = e, =,,, µ {e } = Ù R Õß Ô tr[µ] = e [µ]e = = = (e u 2 dµ(u = dµ(u = µ(, Û tr[µ] = µ( (3 Ð µ Û Â͵ Borel µ Û» u u dµ(u = I,
3 Ì ¾ ¼ Ball-Barthe Ø 39 Õ I Ù Å» µ Û» à x R, Í x 2 = x u 2 dµ(u µ Û ² Dirac µ Û» µ = m c i δ(u i, u i, c i > 0, µ δ Û m c i u i u i = I Ï»ÈÊ [8-0] Ð {x i } R, Ê [x,, x ] ÙÌ»¼ x,, x Ë Ö Ð, 2,, ÈÇ, 2,,, Á τ(, 2,, ÅÖÏßÓ ÊÛ Jese ʱͫР(Ω, µ ÛÈ ϕ ÛÈ µ(ω =, g Û Ω ÂÖ ( ϕ g dµ ϕ g dµ Ω Ω (4a ¼ ϕ ÛÇ ϕ ÛÈ ¼¼ g supp µ ÛÐ ( ϕ g dµ ϕ g dµ Ω Ω (4b ¼ ϕ ÛÇ ¼¼ g supp µ ÛÐ 2 µ» «ß Đ ÑÈ ÅÅ ½½ ÑÈ ÜÈÔ ÔÚ Ç³ Ð Q i, i =,, m, Û Å Q i = Q (i Q (i, Õ Q (i, =,,, Ù i È Ð
320 3 Ð λ i 0, i =,, m, Đ λ Q + + λ m Q m m λ i Q (i ( m det λ i Q i = det m λ i Q (i = i,,i m λ i λ i det Q (i Q (i λ Q + + λ m Q m ÛÏ λ i Åϵ¹ Ð ( m det λ i Q i = i,,i m 2 Ð Q,, Q Û Å D(Q,, Q =! λ i λ i D(Q i,, Q i (i,,i P(,, ÍÑÈ Ä det Õ P(,, Ù,, ÍÇ ĐÙ Q (i Q (i, ² D(Q,, Q Á Ï Q,, Q, ±³Ð (5 Ä 2 ÂÚ ÏÑÈ Ô Ô [] È D(Q,, Q = detq È 2 D(Q i,, Q i = D(Q,, Q, ÍÕ (i,, i P(,, ( m È 3 D Q i,,, m Q i, = m m D(Q i,,, Q i,, µ Q i, i = i = i = i = Û Å i m, È 4 D(t Q,, t Q = t t D(Q,, Q, t,, t 0 2 [4] Ð Q Û Å Û Q ÅÑÈ = 0,,, D (Q = D(Q,, Q, I,, I }{{}}{{} ² ¼ = 0 Ó D 0 (Q = detq; ¼ = Ó D (Q = deti = Ð Q Û Å λ 0, det(q + λi = λ ( =0 k < <k detq k,,k,
3 Ì ¾ ¼ Ball-Barthe Ø 32 Õ Q k,,k Ù Q k,, k Æ» I k,, k «Ð ½ k < < k, = 0,,, Ä Å Q = Q, Q 2 = I, λ =, λ 2 = λ º (5 Õ Q (i det(q + λi = λ i λ i det (6 i,,i 2 Q (i Õ λ il, l =,,, Û±³ λ i λ i Õ l ÈÓ È λ il ÓÏ λ ÓÏ λ i λ i Õ k,, k ÈÓ±³½ λ, Ñ ÈÓÛ, Ó λ i λ i = λ Ó Q (i Q (i Ñ Û Q Õ È Â det(q + λi = λ ( Õ k,, k Û» I k,, k =0 k < <k 2 Ð Q Û Å λ 0, ( det(q + λi = λ D (Q =0 detq k,,k Ä Å Q = Q, Q 2 = I, λ =, λ 2 = λ º (6 Õ det(q + λi = λ i λ i D (Q i,, Q i = = Ìdz dz 2 D (Q = i,,i 2 =0 =0 k < <k ( λ D(Q,, Q, I,, I }{{}}{{} ( λ D (Q det Q k,,k (, = 0,,, ¼ = Ó D (Q = trq (7
322 3 3 [µ]»¾ 3 Ð Q, =,,, Û Å Q = q q, ÍÕ q = (q,, q, qi R, i, D(Q,, Q =! [q,, q ] 2 Ä Ã x = (x,, x R, Í x x = ÌÄ 2 ÂÚ Ä D(Q,, Q =! =! =! =! (,, P(,, (,, P(,, (,, P(,, (,, P(,, q =! det Ð Q = q =! [q,, q ] 2 m i = q i, R, =,,, Q i, = det det x x x x Q ( Q ( q q det q q q q q q ( τ(,, det (,, P(,, m i = D(Q,, Q =! q q ( τ(,, q q ( τ(,, q q q i, q i,, ÍÕ Q i, Û Å m i = m [q i,,, q i,] 2 i =
3 Ì ¾ ¼ Ball-Barthe Ø 323 Ä ÌÔ 3 dz 3, ( m D(Q,, Q = D i = m m Q i,,, m i = Q i, = D(Q i,,, Q i, i = m i = m = D(q i, q i,,, q i, q i, i = i = m m = [q i,,, q i,] 2! i = i = ¼ µ i (i =,, Û ²Ó Ì 2 Ð µ,, µ ² µ Ù supp µ = {u,,, u m,}, =,,, m (i [µ ] = µ (u i, u i, u i,; i = (ii D([µ ],, [µ ] =! m i = m i = [u i,,, u i,] 2 µ (u i, µ (u i, 3 Ð µ,, µ Â͵ Borel (i [µ ] = u u dµ (u, =,, ; (ii D([µ ],, [µ ] =! [u,, u ] 2 dµ (u dµ (u Í D 0 ([µ] = det[µ], (8 ÂÚ D 0 ([µ] = [u,, u ] 2 dµ(u dµ(u (9! ÏÛ µ» [u,, u ] 2 dµ(u dµ(u = (0! 4 Ð ν Â͵ Borel µ» D([ν], [µ],, [µ] = ν(s
324 3 Ä Æ µ» [µ] = I, Ì (3 (7 D([ν], [µ],, [µ] = D([ν], I,, I = D ([ν] = tr[ν] = ν(s 5 Ð µ» [u, u 2,, u ] 2 dµ(u 2 dµ(u =, u (! Ä Ì 4 D([ν], [µ],, [µ] = ν(s ν Û ²Å (Û ν( =, supp ν = {u} º D([ν], [µ],, [µ] = Ì 3 D([ν], [µ],, [µ] = [u, u 2,, u ] 2 dν(udµ 2 (u 2 dµ (u! [u, u 2,, u ] 2 dµ(u 2 dµ(u = (! 4 Ball-Barthe»» ű «Ý± Ê ½ÑÈ ÔÚ Ê± Ball-Barthe È «Ä dν = tdµ, ν Borel [ν] = t(u u u dµ(u ÏÛ Ì (8 (9, ( det t(uu udµ(u = det([ν] = D 0 ([ν] = t(u t(u [u,, u ] 2 dµ(u dµ(u!
3 Ì ¾ ¼ Ball-Barthe Ø 325 Ì (0 [u,, u ] 2 dµ(u dµ(u! ÛÈÅ ÏÛ Ì 3, Jese ÂÚ 5, ( det t(u u u dµ(u = t(u t(u [u,, u ] 2 dµ(u dµ(u! S { } exp log(t(u t(u [u,, u ] 2 dµ(u dµ(u! { ( } = exp log t(u i [u,, u ] 2 dµ(u dµ(u! { ( } = exp log t(u [u,, u ] 2 dµ(u 2 dµ(u dµ(u! { } = exp (! log t(u dµ(u! S { } = exp log t(u dµ(u Ì log Ç Ú Jese (4b ¼¼ u,, u supp µ, t(u i ÛÈ µ d µ = [u,, u ] 2 dµ(u dµ(u! ÛÈÅ Ì [4] Õdz A, ß Ï u,, u supp µ, t(u i ÛÈ Ball-Barthe Á» ÚÍ 2 «Ä Ì 3, AM-GM ÂÚ 5, ( det t(u u u dµ(u = t(u t(u [u,, u ] 2 dµ(u dµ(u! t (u + + t (u [u,, u ] 2 dµ(u dµ(u! S = ( t (u [u,, u ] 2 dµ(u 2 dµ(u dµ(u! = (! t (u dµ(u! = t (u dµ(u
326 3 Ì AM-GM ³Õ ¼¼ t(u = = t(u, u,, u supp µ, Û c > 0, Ö Ã u supp µ, t(u = c  ÊÁ» Ball-Barthe ʱ½ ÖÈ 3 «Ä t(u = ρ K (u, u, dµ = ω ds(u, ʳ 2 ( det ρ K (uu uds(u ρ ω S ω K(udS(u = V (K, S ω Û ( V (K ω det ρ K (uu uds(u, ¼¼ c > 0, Ö Ã u, ρ K (u = c, Û K ÛÈ c Õ [] Mahler K Ei Übertragugsprizip für kovexe körper [J] Casopis Pĕst, Mat Fys, 939, 68: 93-02 [2] Ball K Volume ratios ad a reverse isoperimetric iequality [J] J Lodo Math Soc, 99, 44: 35-359 [3] Barthe F O a reverse form of the Brascamp-Lieb iequality [J] Ivet Math, 998, 34: 335-36 [4] Lutwak E, Yag D, Zhag G Volume iequality for subspaces of Lp [J] J Differetial Geom, 2004, 68: 65-69 [5] Lutwak E, Yag D, Zhag G Volume iequalities for isotropic measures [J] Amer J Math, 2007, 29: 7-723 [6] Lutwak E, Yag D, Zhag G A volume iequality for polar bodies [J] J Differetial Geom, 200, 84: 63-78 [7] Schuster F, Weberdorfer M Volume iequalities for asymmetric wulff shapes [J] J Differetial Geom, 202, 92: 263-283 [8] Giaopoulos A A, Papadimitrakis M Isotropic surface area measures [J] Mathematika, 999, 46: -3 [9] Giaopoulos A A, Milma V D Extremal problems ad isotropic positios of covex bodies [J] Israel J Math, 2000, 7: 29-60 [0] Giaopoulos A A, Milma V D, Rudelso M Covex bodies with miimal mea width [M]//Milma V, Schechtma G Geometric Aspects of Fuctioal Aalysis Heidelberg, Berli: Spriger-Verlag, 2000: 8-93 [] Scheider R Covex Bodies: the Bru-Mikowski Theory [M] 2d ed Cambridge: Cambridge Uiversity Press, 204