A connected compact locally Chebyshev set in a finite-dimensional space is a Chebyshev set

Abstract

Let $X$ be a Banach space. A set $M\subset X$ is a Chebyshev set if, for each $x\in X$, there exists a unique best approximation to $x$ in $M$. A set $M$ is locally Chebyshev if, for any point $x\in M$, there exists a Chebyshev set $F_x\subset M$ such that some neighbourhood of $x$ in $M$ lies in $F_x$. It is shown that each connected compact locally Chebyshev set in a finite-dimensional normed space is a Chebyshev set. Bibliography: 11 titles.