Abstract. A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality, the first of its type, is proved, together with precise equality conditions, and is shown to be the best possible from several points of view. A new Gaussian Brunn-Minkowski inequality is proposed and proved to be true in some significant special cases. Throughout the study attention is paid to precise equality conditions and conditions on the coefficients of dilatation. Interesting links are found to the S-inequality and the (B) conjecture. An example is given to show that convexity is needed in the (B) conjecture.
Abstract. We prove that the unit cube B n ∞ is a strict local minimizer for the Mahler volume product vol n (K)vol n (K * ) in the class of origin symmetric convex bodies endowed with the Banach-Mazur distance.
Abstract. The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of general measures most of which were known before only in the case of the standard Lebesgue measure. We solve an analog of the Busemann-Petty problem for the case of general measures. In addition, we show that there are measures, for which the answer to the generalized Busemann-Petty problem is affirmative in all dimensions. Finally, we apply the latter fact to prove a number of different inequalities concerning the volume of sections of convex symmetric bodies in R n and solve a version of generalized Busemann-Petty problem for sections by k-dimensional subspaces.
Abstract. In this paper we consider the following analog of Bezout inequality for mixed volumes:We show that the above inequality is true when ∆ is an n -dimensional simplex and P1, . . . , Pr are convex bodies in R n . We conjecture that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be an n -dimensional simplex. We prove that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to ∆ ), which confirms the conjecture when ∆ is a simple polytope and in the 2 -dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.
Let us define for a compact set A ⊂ R n the sequence A(k) = a 1 + · · · + a k k : a 1 , . . . , a k ∈ A = 1 k A + · · · + A k times.It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that A(k) approaches the convex hull of A in the Hausdorff distance induced by the Euclidean norm as k goes to ∞. We explore in this survey how exactly A(k) approaches the convex hull of A, and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on R n , the volume deficit (the difference of volumes), a nonconvexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of A(k) does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets A with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence A(k), and both the Hausdorff distance and effective standard deviation are eventually monotone (once k exceeds n). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets (showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex), falsify a conjecture of Dyn and Farkhi (2004), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.
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