Dual space of the Minkowski-Rådström-Hörmander space over $\mathbb{R}^{2}$

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).