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Problem 102: Posed by John Reay

Publisher
Elsevier Science
Year
1989
Tongue
English
Weight
74 KB
Volume
74
Category
Article
ISSN
0012-365X

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โœฆ 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.

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Given integers I-Z and 1p1, how many distinct solutions {a, b, c} are there to the system of equations a+b+c=m a\*+b\*+c\*=n with aabacaO? With the condition on the non-nq$iv, ,jr of aLs b and c removed, the problem was solved in [I]. A solution to the more constrained problem would have application