Work

Extremizers and Stability of the Betke–Weil Inequality

Year: 2022

Type: article

Abstract: Let K be a compact convex domain in the Euclidean plane. The mixed area A(K,−K) of K and −K can be bounded from above by 1/(6 3)L(K)2, where L(K) is the perimeter of K. This was proved by Ulrich Betke and Wolfgang Weil [5]. They also showed that if K is a polygon, then equality holds if and only if K is a regular triangle. We prove that among all convex domains, equality holds only in this case, as conjectured by Betke and Weil. This is achieved by establishing a stronger stability result for the geometric inequality 6 3A(K,−K)≤L(K)2.

Source: The Michigan Mathematical Journal

DOI: 10.1307/mmj/20216063

Authors: Ferenc A. Bartha, Ferenc Bencs, Károly J. Böröczky, Daniel Hug

Institutions: University of Szeged, HUN-REN Alfréd Rényi Institute of Mathematics, Karlsruhe Institute of Technology

Language: English

Topic: Point processes and geometric inequalities

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: Deutsche Forschungsgemeinschaft, Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, National Research, Development and Innovation Office