Convexity of Chebyshev Sets Contained in a Subspace

Convex Chebyshev sets $M$ in a linear space $X$ with norm or nonsymmetric norm $(X,\|\cdot\|)$ which are contained in a subspace $H$ of $X$ are considered. It is proved that if $|\cdot|_{H,\theta}$ is the nonsymmetric norm on $H$ determined by the Minkowski functional of $(B-\theta)\cap H$, where $B$ is the unit ball of $X$ and $\|\theta\|<1$, with respect to 0, then $M$ is a Chebyshev set in $(H,|\cdot|_{H,\theta})$ for any $\theta$. From this result sufficient and necessary conditions for the convexity of Chebyshev sets and bounded Chebyshev sets contained in a subspace $H$ of $X$ are derived.