Eprints of Rényi Institute

The perimeter of rounded convex planar sets

Csirmaz, László (2007) The perimeter of rounded convex planar sets. Periodica Mathematica Hungarica, 54 (1). pp. 31-49.

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Abstract

A convex set is inscribed into a rectangle with sides $a$ and $1/a$ so that the convex set has points on all four sides of the rectangle. By "rounding" we mean the composition of two orthogonal linear transformations parallel to the edges of the rectangle, which makes a unit square out of the rectangle. The transformation also applied to the convex set, which now has the same area, and is inscribed into a square. One would expect this transformation to decrease the perimeter. Interestingly this is not always the case. For each $a$ we determine the largest and smallest possible increase of the perimeter. We also look at the case when the inscribed convex set is a triangle.

Item Type:Article
Keywords and phrases:geometric inequality, planar convex set, perimeter, isoperimetric problem
Subjects:52A10 Convex sets in $2$ dimensions (including convex curves)
51M16 Inequalities and extremum problems
53A05 Surfaces in Euclidean space
26D07 Inequalities involving other types of functions
51M25 Length, area and volume
Divisions:Research Groups > Theoretical cryptography
Last Modified:10 Jul 2009 17:08

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