Problem 103: Posed by Robert A. Melter

• Publisher: Elsevier Science
• Year: 1989
• Tongue: English
• Weight: 74 KB
• Volume: 74
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(89)90151-9

✦ Synopsis

Theorem (See [3] for a proof and definitions). Let S c lPd be Q set of at least (d + l)(r -1) + (k + I) strongly independent points, where 0 6 k G d. Then S has a partition S=SU-US' so that the k-dimensional volume V(k) = ~01&&l conv Si] satisfies V(k) > 0. If (r, k) = (2,0) this is Radon's theorem; (r, k) = (r, 0) is Tvergerg's theorem.

Problem. Find a suitable independence condition for S to assure V(k) > 1. See [l] and [2] for a similar Helly--type results.