Sets with (d − 2)-dimensional kernels

Marilyn Breen

Vol. 77 (1978), No. 1, 51–55

DOI: 10.2140/pjm.1978.77.51

Abstract

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 {K_r} such that every point of T sees each K_r via S, (aff K_r) ∩ S = K_r, and distinct pairs aff K_r either are disjoint or lie in a d-flat containing T. Furthermore, assume that when T is affinely independent, then the corresponding set K_r 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

Milestones

Received: 20 June 1977

Revised: 7 November 1977

Published: 1 July 1978