On points at which a set is cone-shaped

Edelstein, M.; Keener, L.; O’Brien, R.

doi:10.1090/S0002-9939-1977-0454593-6

On points at which a set is cone-shaped
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by M. Edelstein, L. Keener and R. O’Brien PDF

Proc. Amer. Math. Soc. 66 (1977), 327-330 Request permission

Abstract:

A set $\mathcal {S}$ in a normed linear space X is said to be cone-shaped at $x \in X$ if there is a closed half-space that has x in its bounding hyperplane and contains $\{ y \in \mathcal {S}:[x,y] \subset S\}$. The point x is called a cone point. In this paper it is shown that if X has an equivalent uniformly convex and uniformly smooth norm and if $\mathcal {S}$ is a closed bounded subset with the finite visibility property for cone points (i.e., for every finite set F of cone points of S there is a point $z \in S$ such that $[z,y] \subset \mathcal {S}$ for all $y \in F$), then S is starshaped.

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