Infinitesimal form of Brunn-Minkowski type inequalities

Transcription

1 Andrea Colesanti Università di Firenze Infinitesimal form of Brunn-Minkowski type inequalities Convex and Discrete Geometry Vienna, 4-8 July, 2016 Dedicated to professor Peter Gruber on the occasion of his 75th birthday

2 At this point of the conference it is difficult to add any word to what has been said about the relevance as a mathematician, the generosity, and the quality to be a gentlemen in any circumstance, of Peter Gruber. I simply associate myself to the previous speakers, and express my gratitude to Peter Gruber and wish him a hundred of days like this!

3 The results presented in this communication are obtained in collaboration with Galyna Livshyts and Arnaud Marsiglietti.

4 The Brunn-Minkowski inequality

5 The Brunn-Minkowski inequality K n = {convex bodies in R n }; V n = Lebesgue measure.

6 The Brunn-Minkowski inequality K n = {convex bodies in R n }; V n = Lebesgue measure. V n ((1 t)k + tl) 1/n (1 t)v n (K) 1/n + tv n (L) 1/n, (BM) for every K, L K n and t [0, 1].

7 The Brunn-Minkowski inequality K n = {convex bodies in R n }; V n = Lebesgue measure. V n ((1 t)k + tl) 1/n (1 t)v n (K) 1/n + tv n (L) 1/n, (BM) for every K, L K n and t [0, 1]. Equivalently, the functional is concave in K n. G : K n R, K G(K) = V n (K) 1/n

8 The Brunn-Minkowski inequality K n = {convex bodies in R n }; V n = Lebesgue measure. V n ((1 t)k + tl) 1/n (1 t)v n (K) 1/n + tv n (L) 1/n, (BM) for every K, L K n and t [0, 1]. Equivalently, the functional G : K n R, K G(K) = V n (K) 1/n is concave in K n. Hence (heuristically): D 2 G(K) 0, K K n, (IBM)

9 The Brunn-Minkowski inequality K n = {convex bodies in R n }; V n = Lebesgue measure. V n ((1 t)k + tl) 1/n (1 t)v n (K) 1/n + tv n (L) 1/n, (BM) for every K, L K n and t [0, 1]. Equivalently, the functional G : K n R, K G(K) = V n (K) 1/n is concave in K n. Hence (heuristically): D 2 G(K) 0, K K n, (IBM) where D 2 G denotes the second variation of G.

10 The Brunn-Minkowski inequality K n = {convex bodies in R n }; V n = Lebesgue measure. V n ((1 t)k + tl) 1/n (1 t)v n (K) 1/n + tv n (L) 1/n, (BM) for every K, L K n and t [0, 1]. Equivalently, the functional G : K n R, K G(K) = V n (K) 1/n is concave in K n. Hence (heuristically): D 2 G(K) 0, K K n, (IBM) where D 2 G denotes the second variation of G. (IBM) is what we call the infinitesimal form of (BM).

11 What is the second variation of a functional?

12 What is the second variation of a functional? Let G : K n R (sufficiently smooth).

13 What is the second variation of a functional? Let G : K n R (sufficiently smooth). Fix a convex body K, of class C 2,+, and let h be its support function.

14 What is the second variation of a functional? Let G : K n R (sufficiently smooth). Fix a convex body K, of class C 2,+, and let h be its support function. Let φ C (S n 1 ) (test function).

15 What is the second variation of a functional? Let G : K n R (sufficiently smooth). Fix a convex body K, of class C 2,+, and let h be its support function. Let φ C (S n 1 ) (test function). For s [ ɛ, ɛ], and ɛ small enough, h s = h + sφ

16 What is the second variation of a functional? Let G : K n R (sufficiently smooth). Fix a convex body K, of class C 2,+, and let h be its support function. Let φ C (S n 1 ) (test function). For s [ ɛ, ɛ], and ɛ small enough, h s = h + sφ = support function of a convex body K s C 2,+.

17 What is the second variation of a functional? Let G : K n R (sufficiently smooth). Fix a convex body K, of class C 2,+, and let h be its support function. Let φ C (S n 1 ) (test function). For s [ ɛ, ɛ], and ɛ small enough, h s = h + sφ = support function of a convex body K s C 2,+. We set d 2 ds 2 G(K s) s=0

18 What is the second variation of a functional? Let G : K n R (sufficiently smooth). Fix a convex body K, of class C 2,+, and let h be its support function. Let φ C (S n 1 ) (test function). For s [ ɛ, ɛ], and ɛ small enough, h s = h + sφ = support function of a convex body K s C 2,+. We set d 2 ds 2 G(K s) = (D 2 G(K) φ, φ). s=0

19 What is the second variation of a functional? Let G : K n R (sufficiently smooth). Fix a convex body K, of class C 2,+, and let h be its support function. Let φ C (S n 1 ) (test function). For s [ ɛ, ɛ], and ɛ small enough, h s = h + sφ = support function of a convex body K s C 2,+. We set d 2 ds 2 G(K s) = (D 2 G(K) φ, φ). s=0 D 2 G(K) is a quadratic form acting on test functions.

20 (BM) (IBM)

21 (BM) (IBM) Let G = V 1/n n.

22 (BM) (IBM) Let G = V 1/n n. Proposition. The (BM) inequality is equivalent to the condition (D 2 G(K) φ, φ) 0, K C 2,+, φ C (S n 1 ). (IBM)

23 (BM) (IBM) Let G = V 1/n n. Proposition. The (BM) inequality is equivalent to the condition (D 2 G(K) φ, φ) 0, K C 2,+, φ C (S n 1 ). (IBM) All in all, (BM) turns out to be equivalent to a whole class of functional inequalities.

24 (BM) (IBM) Let G = V 1/n n. Proposition. The (BM) inequality is equivalent to the condition (D 2 G(K) φ, φ) 0, K C 2,+, φ C (S n 1 ). (IBM) All in all, (BM) turns out to be equivalent to a whole class of functional inequalities. These are inequalities of Poincaré type on S n 1.

25 The standard Poincare inequality on Sn 1

26 The standard Poincaré inequality on S n 1 S n 1 φ 2 dx 1 n 1 S n 1 φ 2 dx,

27 The standard Poincaré inequality on S n 1 φ 2 dx 1 φ 2 dx, S n 1 n 1 S n 1 φ C (S n 1 ) : φdx = 0. S n 1

28 The standard Poincaré inequality on S n 1 φ 2 dx 1 φ 2 dx, S n 1 n 1 S n 1 φ C (S n 1 ) : φdx = 0. S n 1 1 The constant n 1 is sharp (equality is obtained for restriction of linear functions to S n 1 ).

29 The standard Poincaré inequality on S n 1 φ 2 dx 1 φ 2 dx, S n 1 n 1 S n 1 φ C (S n 1 ) : φdx = 0. S n 1 1 The constant n 1 is sharp (equality is obtained for restriction of linear functions to S n 1 ). An equivalent formulation is S n 1 φ 2 dx 1 S n 1 ( S n 1 φdx ) 2 1 n 1 S n 1 φ 2 dx.

30 More general Poincaré inequalities

31 More general Poincaré inequalities ( ) 2 Aφ 2 dx Bφdx S n 1 S n 1 n 1 S n 1 i,j=1 C ij φ i φ j dx.

32 More general Poincaré inequalities Here: ( ) 2 Aφ 2 dx Bφdx S n 1 S n 1 n 1 S n 1 i,j=1 C ij φ i φ j dx. φ i, i = 1,..., n 1, are the first covariant derivatives of φ.

33 More general Poincaré inequalities Here: ( ) 2 Aφ 2 dx Bφdx S n 1 S n 1 n 1 S n 1 i,j=1 C ij φ i φ j dx. φ i, i = 1,..., n 1, are the first covariant derivatives of φ. A, B are positive (smooth) functions on S n 1.

34 More general Poincaré inequalities Here: ( ) 2 Aφ 2 dx Bφdx S n 1 S n 1 n 1 S n 1 i,j=1 C ij φ i φ j dx. φ i, i = 1,..., n 1, are the first covariant derivatives of φ. A, B are positive (smooth) functions on S n 1. The matrix is positive definite. (C ij ) i,j=1...,n,

35 How (BM) is related to the Poincaré inequality?

36 How (BM) is related to the Poincaré inequality? n = 2;

37 How (BM) is related to the Poincaré inequality? n = 2; K K 2 of class C 2,+,

38 How (BM) is related to the Poincaré inequality? n = 2; K K 2 of class C 2,+, h = support function of K, defined on S 1 [0, 2π).

39 How (BM) is related to the Poincaré inequality? n = 2; K K 2 of class C 2,+, h = support function of K, defined on S 1 [0, 2π). V 2 (K) = 1 2 2π 0 h(h + h )dt.

40 How (BM) is related to the Poincaré inequality? n = 2; K K 2 of class C 2,+, h = support function of K, defined on S 1 [0, 2π). φ C ([0, 2π)); V 2 (K) = 1 2 2π 0 h(h + h )dt.

41 How (BM) is related to the Poincaré inequality? n = 2; K K 2 of class C 2,+, h = support function of K, defined on S 1 [0, 2π). V 2 (K) = 1 2 2π 0 h(h + h )dt. φ C ([0, 2π)); K s K 2 such that its support function is h s := h + sφ (for small s ).

42 How (BM) is related to the Poincaré inequality? n = 2; K K 2 of class C 2,+, h = support function of K, defined on S 1 [0, 2π). V 2 (K) = 1 2 2π 0 h(h + h )dt. φ C ([0, 2π)); K s K 2 such that its support function is h s := h + sφ (for small s ). V 2 (K s ) = 1 2 2π 0 (h + sφ)(h + sφ + (h + sφ ))dt.

43 ( ) D 2 V 1/2 2 (K)φ, φ = d 2 V2 ds 2 (K s ) 0 s=0 (IBM)

44 ( ) D 2 V 1/2 2 (K)φ, φ = d 2 V2 ds 2 (K s ) 0 s=0 d 2 ds 2 2π (h + sφ)(h + sφ + (h + sφ ))dt 0. 0 s=0 (IBM)

45 ( ) D 2 V 1/2 2 (K)φ, φ = d 2 V2 ds 2 (K s ) 0 s=0 d 2 ds 2 After computations: [ 1 [V 2 (K)] 3/2 2 V 2 (K) 2π (h + sφ)(h + sφ + (h + sφ ))dt 0. 0 s=0 2π 0 (IBM) ( 2π ) 2 ] φ(φ + φ)dt φ(h + h )dt 0, 0

46 ( ) D 2 V 1/2 2 (K)φ, φ = d 2 V2 ds 2 (K s ) 0 s=0 d 2 ds 2 After computations: [ 1 [V 2 (K)] 3/2 2 V 2 (K) 2π (h + sφ)(h + sφ + (h + sφ ))dt 0. 0 s=0 2π whence (with an integration by parts) 2π 0 φ 2 dt 0 (IBM) ( 2π ) 2 ] φ(φ + φ)dt φ(h + h )dt 0, 0 ( 1 2π 2 φ(h + h )dx) 2V 2 (K) 0 2π 0 (φ ) 2 dt.

47 Infinitesimal (BM) in dimension 2

48 Infinitesimal (BM) in dimension 2 (D 2 V 1/2 2 (K) φ, φ) 0, φ

49 Infinitesimal (BM) in dimension 2 (D 2 V 1/2 2 (K) φ, φ) 0, φ 2π 0 ( φ 2 1 2π 2 dt φ(h + h )dx) 2V 2 (K) 0 2π 0 (φ ) 2 dt, φ.

50 Infinitesimal (BM) in dimension 2 (D 2 V 1/2 2 (K) φ, φ) 0, φ 2π 0 ( φ 2 1 2π 2 dt φ(h + h )dx) 2V 2 (K) 0 2π 0 (φ ) 2 dt, φ. When K is the unit ball, i.e. h 1, we get precisely the standard Poincaré inequality on S 1 : 2π 0 φ 2 dt 1 ( 2π 2 φdx) 2π 0 2π 0 (φ ) 2 dt.

51 Infinitesimal (BM) in general dimension

52 Infinitesimal (BM) in general dimension Thm. [C. 2008] Let K K n be of class C 2,+. The condition ( ) D 2 Vn 1/n φ, φ 0, φ C (S n 1 ), (IBM)

53 Infinitesimal (BM) in general dimension Thm. [C. 2008] Let K K n be of class C 2,+. The condition ( ) D 2 Vn 1/n φ, φ 0, φ C (S n 1 ), (IBM) is equivalent to tr (c ij )φ 2 dx 1 ( φ det(h ij + hδ ij )dx S n 1 V n (K) S n 1 n 1 c ij φ i φ j dx, φ C (S n 1 ). S n 1 i,j=1 ) 2

54 Infinitesimal (BM) in general dimension Thm. [C. 2008] Let K K n be of class C 2,+. The condition ( ) D 2 Vn 1/n φ, φ 0, φ C (S n 1 ), (IBM) is equivalent to tr (c ij )φ 2 dx 1 ( φ det(h ij + hδ ij )dx S n 1 V n (K) S n 1 n 1 c ij φ i φ j dx, φ C (S n 1 ). S n 1 i,j=1 Here: h ij = second covariant derivatives of h; δ ij= Kronecker symbols; (c ij ) = cofactor matrix of (h ij + hδ ij ). ) 2

55 Infinitesimal (BM) in general dimension Thm. [C. 2008] Let K K n be of class C 2,+. The condition ( ) D 2 Vn 1/n φ, φ 0, φ C (S n 1 ), (IBM) is equivalent to tr (c ij )φ 2 dx 1 ( φ det(h ij + hδ ij )dx S n 1 V n (K) S n 1 n 1 c ij φ i φ j dx, φ C (S n 1 ). S n 1 i,j=1 Here: h ij = second covariant derivatives of h; δ ij= Kronecker symbols; (c ij ) = cofactor matrix of (h ij + hδ ij ). For h 1 (K = unit ball), this is the standard Poincaré inequality on S n 1. ) 2

56 What we did

57 What we did Together with G. Livshyts and A. Marsiglietti, we studied the infinitesimal form of some conjectured inequalities of (BM) type; namely:

58 What we did Together with G. Livshyts and A. Marsiglietti, we studied the infinitesimal form of some conjectured inequalities of (BM) type; namely: the dimensional Brunn-Minkowski inequality for the Gauss measure;

59 What we did Together with G. Livshyts and A. Marsiglietti, we studied the infinitesimal form of some conjectured inequalities of (BM) type; namely: the dimensional Brunn-Minkowski inequality for the Gauss measure; the log-brunn-minkowski inequality.

60 The dimensional (BM) inequality for the Gauss measure

61 The dimensional (BM) inequality for the Gauss measure Let γ n be the Gauss probability measure in R n.

62 The dimensional (BM) inequality for the Gauss measure Let γ n be the Gauss probability measure in R n. Thm. For every K, L K n and for every t [0, 1]: γ n ((1 t)k + tl) γ n (K) 1 t γ n (L) t. (1)

63 The dimensional (BM) inequality for the Gauss measure Let γ n be the Gauss probability measure in R n. Thm. For every K, L K n and for every t [0, 1]: γ n ((1 t)k + tl) γ n (K) 1 t γ n (L) t. (1) Conjecture (Gardner-Zvavitch). For every K, L K n, centrally symmetric, [γ n ((1 t)k + tl)] 1/n (1 t)[γ n (K)] 1/n + t[γ n (L)] 1/n. (2)

64 The dimensional (BM) inequality for the Gauss measure Let γ n be the Gauss probability measure in R n. Thm. For every K, L K n and for every t [0, 1]: γ n ((1 t)k + tl) γ n (K) 1 t γ n (L) t. (1) Conjecture (Gardner-Zvavitch). For every K, L K n, centrally symmetric, [γ n ((1 t)k + tl)] 1/n (1 t)[γ n (K)] 1/n + t[γ n (L)] 1/n. (2) Remarks. (2) is stronger that (1).

65 The dimensional (BM) inequality for the Gauss measure Let γ n be the Gauss probability measure in R n. Thm. For every K, L K n and for every t [0, 1]: γ n ((1 t)k + tl) γ n (K) 1 t γ n (L) t. (1) Conjecture (Gardner-Zvavitch). For every K, L K n, centrally symmetric, [γ n ((1 t)k + tl)] 1/n (1 t)[γ n (K)] 1/n + t[γ n (L)] 1/n. (2) Remarks. (2) is stronger that (1). (2) is false without central symmetry, even if K and L contains the origin (Nayar and Tkocz, 2013).

66 The dimensional (BM) inequality for the Gauss measure Let γ n be the Gauss probability measure in R n. Thm. For every K, L K n and for every t [0, 1]: γ n ((1 t)k + tl) γ n (K) 1 t γ n (L) t. (1) Conjecture (Gardner-Zvavitch). For every K, L K n, centrally symmetric, [γ n ((1 t)k + tl)] 1/n (1 t)[γ n (K)] 1/n + t[γ n (L)] 1/n. (2) Remarks. (2) is stronger that (1). (2) is false without central symmetry, even if K and L contains the origin (Nayar and Tkocz, 2013). Gardner and Zvavitch proved (2) for rectangular boxes. Livshyts, Marsiglietti, Nayar and Zvavitch (2015) proved it in dimension 2, and in general dimension for unconditional convex bodies.

67 The infinitesimal form

68 The infinitesimal form We computed, for an arbitrary K K n of class C 2,+, ( ) D 2 γn 1/n (K)φ, φ, φ C (S n 1 ).

69 The infinitesimal form We computed, for an arbitrary K K n of class C 2,+, ( ) D 2 γn 1/n (K)φ, φ, φ C (S n 1 ). In particular we could prove the following statement. Thm. ( D 2 γ 1/n n ) (B)φ, φ 0, φ C (S n 1 ). where B is any ball centered at the origin.

70 The infinitesimal form We computed, for an arbitrary K K n of class C 2,+, ( ) D 2 γn 1/n (K)φ, φ, φ C (S n 1 ). In particular we could prove the following statement. Thm. ( D 2 γ 1/n n ) (B)φ, φ 0, φ C (S n 1 ). where B is any ball centered at the origin. Note: the perturbation φ does not need to be even.

71 The infinitesimal form We computed, for an arbitrary K K n of class C 2,+, ( ) D 2 γn 1/n (K)φ, φ, φ C (S n 1 ). In particular we could prove the following statement. Thm. ( D 2 γ 1/n n ) (B)φ, φ 0, φ C (S n 1 ). where B is any ball centered at the origin. Note: the perturbation φ does not need to be even. This indicates that the dimensional (BM) inequality for γ n is true close to the unit ball (in an appropriate sense).

72 The log-brunn-minkowski inequality (shortly)

73 The log-brunn-minkowski inequality (shortly) The log-brunn-minkowski inequality was conjectured by Böröczky, Lutwak, Yang and Zhang, who proved it in the two-dimensional case.

74 The log-brunn-minkowski inequality (shortly) The log-brunn-minkowski inequality was conjectured by Böröczky, Lutwak, Yang and Zhang, who proved it in the two-dimensional case. Subsequently, Saroglou established this inequality for unconditional convex bodies in R n.

75 The log-brunn-minkowski inequality (shortly) The log-brunn-minkowski inequality was conjectured by Böröczky, Lutwak, Yang and Zhang, who proved it in the two-dimensional case. Subsequently, Saroglou established this inequality for unconditional convex bodies in R n. As in the previous case, we computed the second variation of the relevant functional at any K K n of class C 2,+,

76 The log-brunn-minkowski inequality (shortly) The log-brunn-minkowski inequality was conjectured by Böröczky, Lutwak, Yang and Zhang, who proved it in the two-dimensional case. Subsequently, Saroglou established this inequality for unconditional convex bodies in R n. As in the previous case, we computed the second variation of the relevant functional at any K K n of class C 2,+, and we checked that it has the right sign (i.e. confirming the conjecture) when K is a ball centered at the origin.

77 The message

78 The message Various (BM) type inequalities are equivalent (or at least related) to a class of Poincaré inequalities on S n 1, which represent their infinitesimal version. This provides, in principle, a way to prove (or confute) them.

79 The message Various (BM) type inequalities are equivalent (or at least related) to a class of Poincaré inequalities on S n 1, which represent their infinitesimal version. This provides, in principle, a way to prove (or confute) them. Nevertheless, it is complicated, in general, to establish the validity of these Poincaré inequalities, and very few cases (typically, corresponding to centered balls) are accessible.

80 The message Various (BM) type inequalities are equivalent (or at least related) to a class of Poincaré inequalities on S n 1, which represent their infinitesimal version. This provides, in principle, a way to prove (or confute) them. Nevertheless, it is complicated, in general, to establish the validity of these Poincaré inequalities, and very few cases (typically, corresponding to centered balls) are accessible. Never-the-nevertheless, very recent results of E. Milman and Kolesnikov, concerning Brunn-Minkowski inequalities on Riemannian manifolds, show that in some cases this approach can be successful.