Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Zero-extreme points and the generalized convex kernel


Author: Arthur G. Sparks
Journal: Proc. Amer. Math. Soc. 67 (1977), 142-146
MSC: Primary 52A10
MathSciNet review: 0461290
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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)\vert x \in {\text{bd}}\;S\} $ and in certain cases, it is known that $ K(1) = \cap \{ K(1,x)\vert 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)\vert 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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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0461290-X
Keywords: Zero-extreme points, extreme points, convex kernel, generalized convex kernel
Article copyright: © Copyright 1977 American Mathematical Society