The perimeter of rounded convex planar sets

Csirmaz László

doi:10.1007/s-10998-007-1031-2

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Periodica Mathematica Hungarica

Volume/Issue:: Volume 54: Issue 1

The perimeter of rounded convex planar sets

Author:

Csirmaz László

Csirmaz László Central European University Budapest Hungary Budapest Hungary

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Pages:: 31–49

Online Publication Date:: 25 Feb 2007

Publication Date:: 01 Mar 2007

Article Category:: Research Article

DOI:: https://doi.org/10.1007/s-10998-007-1031-2

Keywords:: 51M16; 51M25; 26D07; 53A05; 52A10; geometric inequality; planar convex set; perimeter; isoperimetric problem

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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 is 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 of the convex set as well. Interestingly, this is not always the case. For each a we determine the largest and smallest possible increase of the perimeter.

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Periodica Mathematica Hungarica

Print ISSN:: 0031-5303

Online ISSN:: 1588-2829

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Periodica Mathematica Hungarica
Language	English
Size	B5
Year of Foundation	1971
Volumes per Year	2
Issues per Year	4
Founder	Bolyai János Matematikai Társulat - János Bolyai Mathematical Society
Founder's Address	H-1055 Budapest, Hungary Falk Miksa u. 12.I/4.
Publisher	Akadémiai Kiadó Springer Nature Switzerland AG
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245. CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	0031-5303 (Print)
ISSN	1588-2829 (Online)