Zero-extreme points and the generalized convex kernel

Sparks, Arthur G.

doi:10.1090/S0002-9939-1977-0461290-X

Zero-extreme points and the generalized convex kernel
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Proc. Amer. Math. Soc. 67 (1977), 142-146 Request permission

Abstract:

Let S be a compact simply connected set in the plane. Let $K(n)$ denote the generalized convex kernel of S of order n, bd S the boundary of S, $E(0,S)$ the set of 0-extreme points of S, and for $x \in S$, let $K(n,x)$ denote the nth order convex kernel of x in S. It is known that $K(n) = \cap \{ K(n,x)|x \in {\text {bd}}\;S\}$ and in certain cases, it is known that $K(1) = \cap \{ K(1,x)|x \in E(0,S)\}$. The main result of this paper extends the above mentioned results for certain sets. It is shown that $K(n) = \cap \{ K(n,x)|x \in E(0,S)\}$ for certain compact simply connected sets S in the plane. In the process of obtaining this result, a characterization of the boundary is also obtained.

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