Vol. 77, No. 1, 1978

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Sets with (d 2)-dimensional kernels

Marilyn Breen

Vol. 77 (1978), No. 1, 51–55

This work is about the dimension of the kernel of a starshaped set, and the following result is obtained: Let S be a subset of a linear topological space, where S has dimension at least d 2. Assume that for every (d + 1)-member subset T of S there corresponds a collection of (d 2)-dimensional convex sets {Kr} such that every point of T sees each Kr via S, (aff Kr) S = Kr, and distinct pairs aff Kr either are disjoint or lie in a d-flat containing T. Furthermore, assume that when T is affinely independent, then the corresponding set Kr is exactly the kernel of T relative to S. Then S is starshaped and the kernel of S is (d 2)-dimensional.

Mathematical Subject Classification 2000
Primary: 52A30
Secondary: 46A99
Received: 20 June 1977
Revised: 7 November 1977
Published: 1 July 1978
Marilyn Breen