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

  • Ignace I. Kolodner
Conference paper


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.


Quasiconformal Mapping Plane Curve Extremal Property Continuity Argument Opposite Vertex 
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  1. [1]
    Davis, Chandler, An extremal problem for plane convex curves. In Convexity, Proceedings of the 7th symposium of the American Mathematical Society, Providence, R.I., 1973.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1981

Authors and Affiliations

  • Ignace I. Kolodner
    • 1
  1. 1.Department of MathematicsCarnegie-Mellon UniversityPittsburghUSA

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