Results 101 to 110 of about 137 (128)
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Dual Brunn–Minkowski inequality for volume differences
Geometriae Dedicata, 2009The author proves several dual Brunn-Minkowski type inequalities for the volume difference, dual quermassintegral difference, width-integral difference or dual mixed volume difference of star bodies. For instance, it is shown that if \(K,L,M,M'\) are star bodies in \({\mathbb R}^n\) such that \(M'\) is a dilation of \(M\) and \(K\subset M\), \(L\subset
Songjun Lv, Lv Songjun
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The Dual $$\phi $$-Brunn–Minkowski Inequality
Mediterranean Journal of Mathematics, 2021Let \(K, L\subset\mathbb{R}^n\) be star bodies (i.e., star-shaped sets with respect to the origin, having continuous radial function), and let \(\rho_K\) and \(\rho_L\) be the radial functions of \(K\) and \(L\), respectively. Let \(\phi\in C^1((0,\infty))\) be strictly decreasing and concave, and assume that \(\log\circ\phi^{-1}\) is a convex function.
Shi, Wei, Li, Tian, Wang, Weidong
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On Discrete LOG-Brunn--Minkowski Type Inequalities
SIAM Journal on Discrete Mathematics, 2022zbMATH Open Web Interface contents unavailable due to conflicting licenses.
María A. Hernández Cifre +1 more
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A Brunn–Minkowski-Type Inequality
Geometriae Dedicata, 1999For convex bodies \(K,L\) in \(\mathbb{R}^n\), let \(M(K,L): =\max_{x\in \mathbb{R}^n}|K\cap(x+L)|\) (where \(|\cdot|\) denotes volume). The author conjectures that \[ |K+L |^{1/n}\geq M(K,L)^{1/n} +{|K |^{1/n} |L|^{1/n}\over M(K,L)^{1/n}}, \] which would be a useful improvement of the Brunn-Minkowski theorem.
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2000
This section is basic for our further considerations and is devoted to those convex sets which lie in finite-dimensional topological vector spaces. As mentioned in the previous section, if E is an arbitrary finite-dimensional (Hausdorff) topological vector space, then E is isomorphic to some Euclidean space R n .
V. V. Buldygin, A. B. Kharazishvili
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This section is basic for our further considerations and is devoted to those convex sets which lie in finite-dimensional topological vector spaces. As mentioned in the previous section, if E is an arbitrary finite-dimensional (Hausdorff) topological vector space, then E is isomorphic to some Euclidean space R n .
V. V. Buldygin, A. B. Kharazishvili
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More Generalizations of Hartfiel’s Inequality and the Brunn–Minkowski Inequality
Bulletin of the Iranian Mathematical Society, 2020Let \(A,B\in\mathbb{C}^{n\times n}\) be positive definite. Minkowski's determinant inequality (also known as the matrix form of the Brunn-Minkowski inequality) states that \[ (\det{(A+B)})^\frac{1}{n}\ge(\det{A})^\frac{1}{n}+(\det{B})^\frac{1}{n}. \] It has several refinements and generalizations. Some of them are extended to sector matrices. A matrix \
Dong, Sheng, Wang, Qing-Wen
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The Brunn–Minkowski Inequality and Nonconvex Sets
Geometriae Dedicata, 1997The author aims at improving the Brunn-Minkowski inequality for nonconvex sets, by introducing the convex hull of one of the sets into the estimate. For a Borel set \(B\) in \(\mathbb{R}^d\), define \[ \xi(a): =\inf\bigl\{\mu (A+B):A \text{ Borel set, } \mu(A)=a \bigr\}, \] where \(\mu\) denotes Lebesgue measure, and let \(\mu(B)=b\), \(\mu(\text{conv}
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Companions to the Brunn–Minkowski inequality
PositivityLet \(\mathcal{C}\) be the family of all compact convex sets in \(\mathbb{R}^n\). For \(A, B \in \mathcal{C}\) denote by \(\operatorname{Vol}(A)\) the Lebesque measure of \(A\) and by \(\Delta (A,B) = \big[\operatorname{Vol}(A)\big]^{1/n} + \big[\operatorname{Vol}(B)\big]^{1/n} - \big[\operatorname{Vol}(A + B)\big]^{1/n}\). An \((m + 1)\)-tuple \((B_0,
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On the similarity of the entropy power inequality and the Brunn- Minkowski inequality (Corresp.)
IEEE Transactions on Information Theory, 1984Summary: The entropy power inequality states that the effective variance (entropy power) of the sum of two independent random variables is greater than the sum of their effective variances. The Brunn-Minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii. Both these inequalities
Max H. M. Costa, Thomas M. Cover
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On Brunn–Minkowski-Type Inequalities for Polar Bodies
The Journal of Geometric Analysis, 2014Let \( {\mathcal K}^{n}_0\) be the set of all convex bodies in \( {\mathbb R}^n \) containing the origin as an interior point. Given \( K, L \in {\mathcal K}^{n}_0 \), \( 1 \leq p \leq \infty \), and \( \lambda, \mu \geq 0 \), we denote by \( \lambda \cdot K +_p \, \mu \cdot L \) their Firey linear combination, its support function is defined by \[ h (\
Hernández Cifre, María A. +1 more
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