combinatorics / Combinatorial geometry
Erdős, Herzog and Piranian (1958) asked whether the regular n-gon maximizes the product of pairwise distances among n points of fixed diameter. After the recent discovery that it does not for even n, this paper proves the first exponential improvement over the n-gon's value, via a vector-field technique.
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An improved lower bound on the maximal product; the sharp maximizer for the 1958 question remains unknown.
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Erdős, Herzog and Piranian (1958) asked whether the regular n-gon maximizes the product of pairwise distances among n points of fixed diameter. After the recent discovery that it does not for even n, this paper proves the first exponential improvement over the n-gon's value, via a vector-field technique.
An improved lower bound on the maximal product; the sharp maximizer for the 1958 question remains unknown.
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