In this paper we study a quantitative isoperimetric inequality in the plane, related to the isoperimetric deficit \$d\$ and the barycentric asymmetry \$L_0\$. Our aim is to prove that there exists an absolute constant C such that for every planar (convex or compact and connected) set it holds: \$L_0^2\le C d\$. This generalizes some results obtained by B. Fuglede 1993 in [12]. For that purpose, we consider a shape optimization problem in which we minimize the ratio \$d/L_0^2\$both in the class of compact connected sets and in the class of convex sets.

On the quantitative isoperimetric inequality in the plane with the barycentric asymmetry / Bianchini, Chiara; Croce, Gisella. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 2036-2145. - ELETTRONICO. - (2022), pp. 0-0. [10.2422/2036-2145.202107_014]

### On the quantitative isoperimetric inequality in the plane with the barycentric asymmetry

#### Abstract

In this paper we study a quantitative isoperimetric inequality in the plane, related to the isoperimetric deficit \$d\$ and the barycentric asymmetry \$L_0\$. Our aim is to prove that there exists an absolute constant C such that for every planar (convex or compact and connected) set it holds: \$L_0^2\le C d\$. This generalizes some results obtained by B. Fuglede 1993 in [12]. For that purpose, we consider a shape optimization problem in which we minimize the ratio \$d/L_0^2\$both in the class of compact connected sets and in the class of convex sets.
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Bianchini, Chiara; Croce, Gisella
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Utilizza questo identificatore per citare o creare un link a questa risorsa: `https://hdl.handle.net/2158/1283605`