We study, via continuous selections of multivalued maps, the problem of finding a right inverse to the restriction of a linear map to a convex body. Let $M$ be a convex body in a finite dimensional real space $X$ and let $L:X\rightarrow Y$ a (continuous) linear surjection, where $Y$ is also a real vector space. We address here the problem of finding a continuous function $s:L(M)\rightarrow M$ such that $L(s(x))=x$ for every $x\in L(M)$ (i.e. a right inverse of the restriction $L:M\rightarrow L(M)$). This has interesting applications: for example, when $M=A\times B$ with $A$ and $B$ convex body in $\R^n$ and $L$ the sum in $\R^n$, can we find $s=s_1,s_2:A+B\rightarrow A,B$ with $s_1(x)+s_2(x)=x$? \\We try to partially solve these kind of problem using convex closed multivalued mappings and in particular Michael theorem.
Right Inverses of linear maps on convex sets
DE MARCO, GIUSEPPE
2009
Abstract
We study, via continuous selections of multivalued maps, the problem of finding a right inverse to the restriction of a linear map to a convex body. Let $M$ be a convex body in a finite dimensional real space $X$ and let $L:X\rightarrow Y$ a (continuous) linear surjection, where $Y$ is also a real vector space. We address here the problem of finding a continuous function $s:L(M)\rightarrow M$ such that $L(s(x))=x$ for every $x\in L(M)$ (i.e. a right inverse of the restriction $L:M\rightarrow L(M)$). This has interesting applications: for example, when $M=A\times B$ with $A$ and $B$ convex body in $\R^n$ and $L$ the sum in $\R^n$, can we find $s=s_1,s_2:A+B\rightarrow A,B$ with $s_1(x)+s_2(x)=x$? \\We try to partially solve these kind of problem using convex closed multivalued mappings and in particular Michael theorem.Pubblicazioni consigliate
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