Zero-extreme points and the generalized convex kernel

Author:
Arthur G. Sparks

Journal:
Proc. Amer. Math. Soc. **67** (1977), 142-146

MSC:
Primary 52A10

DOI:
https://doi.org/10.1090/S0002-9939-1977-0461290-X

MathSciNet review:
0461290

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Abstract | References | Similar Articles | Additional Information

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 (30:1448)****[2]**J. 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 (38:6459)****[3]**M. H. A. Newman,*Elements of the topology of plane sets of points*, Cambridge Univ. Press, London and New York, 1961, pp. 115-119. MR**0132534 (24:A2374)****[4]**A. G. Sparks,*Intersections of maximal**sets*, Proc. Amer. Math. Soc.**24**(1970), 245-250. MR**0253153 (40:6368)**

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Additional Information

DOI:
https://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