Cliff Smyth, Rutgers University
November 14, 4:30 PM, Rutgers Univ. Hill CORE building, Room 431
Abstract.
A subset S of a metric space X is equilateral if each pair of
points of S determines the same distance. Let e(X) be the maximum
size of an equilateral subset of X.
Let L^p(n) be n-dimensional real space with the metric determined
by the L^p norm. It is known that e(L^2(n)) = n+1 and that n+1
e(L^p(n)) is between n+1 and 2^n. Kusner conjectured in 1983 that e(L^p(n))
= n+1 for all finite p>1. We'll show e(L^p(n)) =
O(n^{(p+1)/(p-1)}). The proof generalizes to give polynomial upper
bounds on the sizes of k-distance sets in L^p(n).