Results 131 to 140 of about 489,827 (153)

More Generalizations of Hartfiel’s Inequality and the Brunn–Minkowski Inequality

Bulletin of the Iranian Mathematical Society, 2020
Let \(A,B\in\mathbb{C}^{n\times n}\) 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 \
Qing-Wen Wang
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The Dual $$\phi $$-Brunn–Minkowski Inequality

Mediterranean Journal of Mathematics, 2021
Let \(K, L\subset\mathbb{R}^n\) 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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A Brunn–Minkowski-Type Inequality

Geometriae Dedicata, 1999
For 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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Brunn-Minkowski inequality

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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The Brunn–Minkowski Inequality and Nonconvex Sets

Geometriae Dedicata, 1997
The 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

Positivity
Let \(\mathcal{C}\) 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, 1984
Summary: 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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Brunn-Minkowski inequality for multiplicities

Inventiones Mathematicae, 1996
Let a connected reductive group \(G\) act in a vector space \(V\). Suppose \(X\) is a closed \(G\)-stable irreducible subvariety of \(\mathbb P(V)\). Let \(F[X] =\bigoplus_mF[X]_m\) be the homogeneous coordinate ring of \(X\). Consider the decomposition of \(F[X]_m\) as \(G\)-module \(F[X]_m =\bigoplus_{\lambda}\mu_m(\lambda)V^\lambda\), where \(V ...
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Inequalities of the Brunn–Minkowski type for Gaussian measures

Probability Theory and Related Fields, 2007
The main purpose of this paper is to study inequalities of the Brunn-Minkowski type for Gaussian measure when the number of sets involved is more than two. Let \(m\geq 2\) be an integer, let \(\gamma\) be the standard Gaussian measure on \(\mathbb{R}^n\), and let \[ \Phi(t)= \int^t_{-\infty}\exp(- s^2/2)\,dx/\sqrt{2\pi},\;-\infty ...
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A direct proof of the Brunn-Minkowski inequality in nilpotent Lie groups

Journal of Mathematical Analysis and Applications, 2022
Julian Pozuelo
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