An Extremal Property of Plane Convex Curves—P. Ungar’s Conjecture

Abstract

Let L be a simple closed, rectifiable, plane curve of perimeter 4p. Using a continuity argument, one can prove that there exist on L four consecutive points, A, A′, B, and B′ which divide the perimeter in four equal parts while the segments AB and A′B′ are orthogonal. In March 1956, according to my recollection, Peter Ungar of CIMS (then the NYU Institute for Mathematics and Mechanics) conjectured, while studying properties of quasiconformal mappings, that: If L is convex, then

$$ \overline {AB} \; + \;\overline {A\prime B\prime } \; \ge \;{\rm{2}}p, $$

with equality iff L is a rectangle.

Research partially supported by NSF Grant MCS76-07567.