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Register a new userA norm - inequality related to affine regular hexagons
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Reinhard Wolf
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Let $(E, lVert . rVert)$ be a two-dimensional real normed space with unit sphere $S = {x in E, lVert x rVert = 1}$. The main result of this paper is the following: Consider an affine regular hexagon with vertex set $H = {pm v_1, pm v_2, pm v_3} subseteq S$ inscribed to $S$. Then we have $$min_i max_{x in S}{lVert x - v_i rVert + lVert x + v_i rVert} leq 3.$$ From this result we obtain $$min_{y in S} max_{x in S}{lVert x - y rVert + lVert x + y rVert} leq 3,$$ and equality if and only if $S$ is a parallelogram or an affine regular hexagon.
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Let $(Omega,{mathcal F},P)$ be a probability space and $L^0({mathcal F})$ the algebra of equivalence classes of real-valued random variables defined on $(Omega,{mathcal F},P)$. A left module $M$ over the algebra $L^0({mathcal F})$(briefly, an $L^0({mathcal F})$-module) is said to be regular if $x=y$ for any given two elements $x$ and $y$ in $M$ such that there exists a countable partition ${A_n,nin mathbb N}$ of $Omega$ to $mathcal F$ such that ${tilde I}_{A_n}cdot x={tilde I}_{A_n}cdot y$ for each $nin mathbb N$, where $I_{A_n}$ is the characteristic function of $A_n$ and ${tilde I}_{A_n}$ its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular $L^0({mathcal F})$-modules: let $V$ and $V^prime$ be two regular $L^0({mathcal F})$-modules such that $V$ contains a free $L^0({mathcal F})$-submodule of rank $2$, if $T:Vto V^prime$ is stable and invertible and maps each $L^0$-line segment onto an $L^0$-line segment, then $T$ must be $L^0$-affine.
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Yuan and Leng (2007) gave a generalization of Ky Fans determinantal inequality, which is a celebrated refinement of the fundamental Brunn-Minkowski inequality $(det (A+B))^{1/n} ge (det A)^{1/n} +(det B)^{1/n}$, where $A$ and $B$ are positive semidefinite matrices. In this note, we first give an extension of Yuan-Lengs result to multiple positive definite matrices, and then we further extend the result to a larger class of matrices whose numerical ranges are contained in a sector. Our result improves a recent result of Liu [Linear Algebra Appl. 508 (2016) 206--213].
We prove that for $alpha in (d-1,d]$, one has the trace inequality begin{align*} int_{mathbb{R}^d} |I_alpha F| ;d u leq C |F|(mathbb{R}^d)| u|_{mathcal{M}^{d-alpha}(mathbb{R}^d)} end{align*} for all solenoidal vector measures $F$, i.e., $Fin M_b(mathbb{R}^d,mathbb{R}^d)$ and $operatorname{div}F=0$. Here $I_alpha$ denotes the Riesz potential of order $alpha$ and $mathcal M^{d-alpha}(mathbb{R}^d)$ the Morrey space of $(d-alpha)$-dimensional measures on $mathbb{R}^d$.
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