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-dimensional, p a V p ! C p B B B = ∈ K K = X with ⊂ = 2 sasmerccne oyin body convex symmetric a is ◦ A Γ steui alof ball unit the is || ) r and , ( X n h ouertoi tleast at is ratio volume the and , x ≤ ( b x † || Vol V X + × ∈ A p ( ∗ Y Vol ×{ X + 1 Y B A ( yteinrproduct inner the by ) n A , Thus, . r w etrsae,let spaces, vector two are p × + || . n and )( 0 F and / x k = B } Y p ) || Vol / )( eteuiu ellipsoid unique the be + A and B ∞ p p to  Vol B . , | ∩ )If .) s oeta h re- the that Note . ∩ | B Y p p { + and , ) B + 0 B , n interpret and |·|| · || ∞ A }× ◦ eteconvex the be | , ℓ ) t sasymmet- a is ≤ . | ∩ p p K Y t × ∞ om and norms r If . ≤ and , uhthat such of ≤ o sets, for 1 A X h· ,be- 4, A } |·|| · || , , is × ·i and × Y Y n F V X K ∞ e - , . , 2 and V ∗ for the rest of the proof, and we can assume to avoid and that confusion that the volume elements on V and V ∗ are equal. Consider the convex body S(K K ) V V , where S is ×2 ◦ ⊂ × ∗ 1 1 the linear given by S(x,y) = (x,x + y). Observe that F K K◦ E E = E . √2 ⊇ ∩2 ⊇ 1 ∩2 1 √2 1

V S(K K◦)= K K◦ ∩ ×2 ∩2 The first inclusion follows from the observation that and that

P S K K K K 2 V,V ( ( 2 ◦)) = +2 ◦. x F = x,x < x x , ∗ × || || h iF || ||K|| ||K◦ Thus: which implies that n (Vol K)(Vol K◦)= (Vol S(K K◦)) n/2 ×2 2 2 2  n  x + x 2 x . K K◦ F n/2 || || || || ≥ || || > 2n (Vol K 2 K◦)(Vol K +2 K◦) n  ∩ n The theorem follows by induction. > 2− (Vol K K◦)(Vol (K K◦)◦).  ∩2 ∩2 The first inequality follows from an estimate of Rogers and Shepard: If C is a symmetric convex body in X Y, where X and Y are vector spaces, then C is at least as big× as (C X) 1 Acknowledgments P (C). (Proof: For all x P (C), C (x + X) contains∩ × a X,Y ∈ X,Y ∩ translate of (C X)(1 x P (C)).) ∩ − || || X,Y ´ Finally, observe that I would like to thank the Institut des Hautes Etudes Scien- tifiques for their hospitality during my stay there. I would also Vol F Vol E like to thank Sean Bates for his encouragementand interest in = 2 this work. Vol E1 sVol E1

[1] and Vitaly D. Milman, New volume ratio proper- riques, Initiation seminar on analysis, 20th year: 1980/1981 ties for convex symmetric bodies in Rn, Invent. Math. 88 (1987), (G. Choquet, M. Rogalsky, and J. Saint-Raymond, eds.), Exp. 319–340. No. 11, 25, Univ. Paris VI, Paris, 1981. [2] Gilles Pisier, The volume of convex bodies and [4] Luis A. Santal´o, Un invariante afin para los cuerpos convexos geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge del espacio de n dimensiones, Portugaliae Math. 8 (1949), 155– University Press, 1989. 161. [3] Jean Saint-Raymond, Sur le volume des corps convexes sym´et-