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March 2013 Dual space of the Minkowski-Rådström-Hörmander space over $\mathbb{R}^{2}$
Jerzy Grzybowski, Ryszard Urbański
Funct. Approx. Comment. Math. 50(1): 199-206 (March 2013). DOI: 10.7169/facm/2014.50.1.9

Abstract

Let $X$ be a real normed vector space and $\mathcal{B}$$(X)$ be the cone of all nonempty bounded closed convex subsets of $X$. For $A,B,C,D\in \mathcal{B}$$(X)$ we have a relation of equivalence defined by $(A,B)\sim (C,D)$ if and only if $\overline{A+D}=\overline{B+C}$. By $[A,B]$ we denote the quotient class of $(A,B)$. The quotient space $\widetilde{X}=\mathcal{B}$$^{2}(X)/_{\sim} $ is a vector space called the Minkowski-Rådström-Hörmander space over $X$. For $\tilde{x}=[A,B]\in \widetilde{X}$ we have the Hausdorff norm $\|\tilde{x}\|_{H}=d_{H}(A,B)=\inf \{\varepsilon >0|\, A\subset B+\varepsilon \mathbb{B},B\subset A+\varepsilon \mathbb{B}\}$ where $\mathbb{B}$ is the closed unit ball in $X$. We also define Bartels-Pallaschke norm $\|\tilde{x}\|_{BP}=\inf \{\|C\|+\|D\|\,|\,(C,D)\in [A,B] \}$, where $\|A\|=\sup_{a\in A}\|a\|$. In this paper we prove that the bilinear function $(\cdot,\cdot):(\widetilde{\mathbb{R}^{2}},\|\cdot\|_{H})\times (\widetilde{\mathbb{R}^{2}},\|\cdot\|_{BP}) \longrightarrow \mathbb{R}$ defined by $(\tilde{x},\tilde{y})=2V(\tilde{x},\tilde{y})+\langle s\tilde{x},s\tilde{y}\rangle$, where $V(\tilde{x},\tilde{y})$ is a generalized mixed volume and $s\tilde{x}$ is a generalized Steiner's point, satisfies the inequality $|(\tilde{x},\tilde{y})|\leq (2\pi +1)\|\tilde{x}\|_{H}\|\tilde{y}\|_{BP}$. We also prove that this bilinear function defines an isomorphic mapping between Banach spaces $(\widetilde{\mathbb{R}^{2}},\|\cdot \|_{BP})$ and the dual space to $(\widetilde{\mathbb{R}^{2}},\|\cdot \|_H)$ (Theorem 2).

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Jerzy Grzybowski. Ryszard Urbański. "Dual space of the Minkowski-Rådström-Hörmander space over $\mathbb{R}^{2}$." Funct. Approx. Comment. Math. 50 (1) 199 - 206, March 2013. https://doi.org/10.7169/facm/2014.50.1.9

Information

Published: March 2013
First available in Project Euclid: 27 March 2014

zbMATH: 1306.46022
MathSciNet: MR3189509
Digital Object Identifier: 10.7169/facm/2014.50.1.9

Subjects:
Primary: 46B20
Secondary: 52A05 , 54B20

Keywords: extreme points , Minkowski-Rådström-Hörmander spaces , pairs of closed bounded convex sets

Rights: Copyright © 2014 Adam Mickiewicz University

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Vol.50 • No. 1 • March 2013
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