Volumes of subset Minkowski sums and the Lyusternik region (original) (raw)
Related papers
The convexification effect of Minkowski summation
EMS Surveys in Mathematical Sciences, 2018
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.
On the volume of the Minkowski sum of line sets and the entropy-power inequality
IEEE Transactions on Information Theory, 1998
We derive a Brunn-Minkowski-type inequality regarding the volume of the Minkowski sum of degenerate sets, namely, line sets. Let A 1 : : : A n be one dimensional sets of unit length, and v 1 : : : v n vectors in R d , d n. Consider the Minkowski sum of the line sets A i v i = fxv i : x 2 A i g i = 1; : : : ; n. We show that the volume of this set sum satis es V d n X i=1
On Minkowski's inequality and its application
Journal of Inequalities and Applications, 2011
In the paper, we first give an improvement of Minkowski integral inequality. As an application, we get new Brunn-Minkowski-type inequalities for dual mixed volumes.
On the monotonicity of Minkowski sums towards convexity
2017
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. By a theorem of Shapley, Folkman and Starr (1969), A(k) approaches the convex hull of A in Hausdorff distance as k goes to ∞. Bobkov, Madiman and Wang (2011) conjectured that Vol n (A(k)) is non-decreasing in k, where Vol n denotes the n-dimensional Lebesgue measure, or in other words, that when one has convergence in the Shapley-Folkman-Starr theorem in terms of a volume deficit, then this convergence is actually monotone. We prove that this conjecture holds true in dimension 1 but fails in dimension n ≥ 12. We also discuss some related 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. Then we consider whether one can have monotonicity in the Shapley-Folkman-Starr theorem when measured using alternate measures of non-convexity, including the Hausdorff distance, effective standard deviation or inner radius, and a non-convexity index of Schneider. For these other measures, we present several positive results, including a strong monotonicity of Schneider's index in general dimension, and eventual monotonicity of the Hausdorff distance and effective standard deviation. Along the way, we clarify the interrelationships between these various notions of non-convexity, demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.
On packing of Minkowski balls. II
arXiv (Cornell University), 2023
This is the continuation of the author's ArXiv presentation ''On packing of Minkowski balls. I". In section 2 we investigate lattice packings of Minkowski balls and domains. By results of the proof of Minkowski conjecture about the critical determinant we devide the balls and domains on 3 classes: Minkowski, Davis and Chebyshev-Cohn. The optimal lattice packings of the balls and domains are obtained. The minimum areas of hexagons inscribed in the balls and domains and circumscribed around their are given. Direct limits of direct systems of Minkowski balls and domains and their critical lattices are calculated.
Proceedings of the Bulgarian Academy of Sciences
We investigate lattice packings of Minkowski balls. By the results of the proof of Minkowski conjecture about the critical determinant we divide Minkowski balls into 3 classes: Minkowski balls, Davis balls and Chebyshev–Cohn balls. We investigate lattice packings of these balls on planes with varying Minkowski metric and search among these packings the optimal packings. In this paper we prove that the optimal lattice packing of the Minkowski, Davis, and Chebyshev–Cohn balls is realized with respect to the sublattices of index two of the critical lattices of corresponding balls.
Minkowski sums of Cantor-type sets
Colloquium Mathematicum, 2010
The classical Steinhaus theorem on the Minkowski sum of the Cantor set is generalized to a large class of fractals determined by Hutchinson-type operators. Numerous examples illustrating the results obtained and an application to t-convex functions are presented.
On the measure of the one-skeleton of the sum of convex compact sets
Journal of the Australian Mathematical Society, 1987
For any two compact convex sets in a Euclidean space, the relation between the volume of the sum of the two sets and the volume of each of them is given by the Brunn-Minkowski inequality. In this note we prove an analogous relation for the one-dimensional Hausdorff measure of the one-skeleton of the above sets. Also, some counterexamples are given which show that the above results are the best possible in some special cases.