k-Dimensional intersections of convex sets and convex kernels

✍ Scribed by Marilyn Breen

• Publisher: Elsevier Science
• Year: 1981
• Tongue: English
• Weight: 328 KB
• Volume: 36
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(81)80019-2

✦ Synopsis

This paper will investigate the dimension of the kernel of a compact starshaped set, and the following result will be obtained:

Let S be a compact set in some linear topological space L. Then for a k with 1 G k G 11, dim ker S 2 k if and only if for some E > 0 and some n-dimensional flat F in L, every f(n. k) points of S see via S a common k-dimensional e-neighborhood in F. If k = 1 or if k = n, the result is best possible. Furthermore, the proof will yield a Helly-type theorem for the dimension of intersections of compact convex sets in W.