Vol. 40, No. 2, 1972

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On extremal figures admissible relative to rectangular lattices

Murray Silver

Vol. 40 (1972), No. 2, 451–457
Abstract

A theorem of Bender states that if a convex figure F contains no point of the two dimensional lattice G, where G is generated by the vectors V 1 and V 2 having enclosed angle 𝜃, then A(F) 12P(F)max(|V 1|,|V 2|sin𝜃) where |V 1||V 2|. In this paper, two questions are answered: (1) Among all convex figures of perimeter L which are admissible relative to a rectangular lattice G, which encloses the maximum area? (2) Can the constant 1/2 in Bender’s theorem be improved? By using the result of (1), the “sharpest possible” inequality of the Bender type is found.

Mathematical Subject Classification 2000
Primary: 52A40
Milestones
Received: 4 February 1970
Revised: 26 April 1971
Published: 1 February 1972
Authors
Murray Silver