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 denote the generalized convex kernel of *S* of order *n*, bd *S* the boundary of *S*, the set of 0-extreme points of *S*, and for , let denote the *n*th order convex kernel of *x* in *S*. It is known that and in certain cases, it is known that . The main result of this paper extends the above mentioned results for certain sets. It is shown that 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.

**[1]**A. M. Bruckner and J. B. Bruckner,*Generalized convex kernels*, Israel J. Math.**2**(1964), 27–32. MR**0171217****[2]**John W. Kenelly, W. R. Hare Jr., B. D. Evans, and W. H. Ludescher,*Convex components, extreme points, and the convex kernel*, Proc. Amer. Math. Soc.**21**(1969), 83–87. MR**0238183**, 10.1090/S0002-9939-1969-0238183-1**[3]**M. H. A. Newman,*Elements of the topology of plane sets of points*, Second edition, reprinted, Cambridge University Press, New York, 1961. MR**0132534****[4]**Arthur G. Sparks,*Intersections of maximal 𝐿_{𝑛} sets*, Proc. Amer. Math. Soc.**24**(1970), 245–250. MR**0253153**, 10.1090/S0002-9939-1970-0253153-3

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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