A generalization of the Blaschke-Lebesgue problem to a kind of convex domains

Shengliang  Pan; Deyan  Zhang; Zhongjun  Chao

doi:10.3934/dcdsb.2016012

Article Contents

2016, Volume 21, Issue 5: 1587-1601. Doi: 10.3934/dcdsb.2016012

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A generalization of the Blaschke-Lebesgue problem to a kind of convex domains

1.
Department of Mathematics, Tongji University, Shanghai 200092
2.
School of Mathematical Sciences, Huaibei Normal University, Huaibei 235000, Anhui Province
3.
Chengdu No.7 High School, Chengdu 610041, Sichuan Province

Received: September 30, 2013

Revised: February 28, 2014

Published: June 2016

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Abstract

In this paper we will introduce for a convex domain $K$ in the Euclidean plane a function $\Omega_{n}(K, \theta)$ which is called by us the biwidth of $K$, and then try to find out the least area convex domain with constant biwidth $\Lambda$ among all convex domains with the same constant biwidth. When $n$ is an odd integer, it is proved that our problem is just that of Blaschke-Lebesgue, and when $n$ is an even number, we give a lower bound of the area of such constant biwidth domains.

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Mathematics Subject Classification: Primary: 52A38, 52A15; Secondary: 52A40.

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