부산대학교 도서관

A set $M\neq\varnothing$ in a metric space $(X,d)$ is said to be $d$-convex if the condition $x,y,z\in X$, $x,y\in M$, $d(x,z)+d(z,y)=d(x,y)$ implies that $z\in M$. Fifteen theorems about $d$-convex sets in a real normed space $R^n$ of dimension $n$ are presented, namely, necessary and sufficient conditions for $d$-convexity of a halfspace $P\subset R^n$ and of a subspace $L\subset R^n$, the expression of $R^n$ as the sum of $n$ 1-dimensional subspaces, the existence of a $d$-convex supporting hyperplane to $K\subset R^n$ if $\text{int}\,K\neq\varnothing$, $\text{bd}\,K\neq\varnothing$, and a condition for $\text{int}\,K$ and $\text{int}\,K\cup\text{bd}\,K$ to be $d$-convex are given. For arbitrary $x\in K$, $K\subset R^n$, the face of $K$ corresponding to $x$ is defined and its $d$-convexity proved. $R^n$ is called a $d$-space if each bounded set in $R^n$ has its $d$-convex hull also bounded. Some theorems about $d$-spaces are derived. Finally, a special orthogonal basis in $R^n$ is discussed.