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Journal of Convex Analysis 20 (2013), No. 1, 157--180
Copyright Heldermann Verlag 2013

The Relative Isoperimetric Inequality: the Anisotropic Case

Francesco Della Pietra
UniversitÓ del Molise, Dipartimento S.A.V.A., FacoltÓ di Ingegneria, Via Duca degli Abruzzi, 86039 Termoli (CB), Italy

Nunzia Gavitone
UniversitÓ di Napoli, Dip. di Matematica e Applicazioni, 80126 Napoli, Italy


We prove a relative isoperimetric inequality in the plane, when the perimeter is defined with respect to a convex, positively homogeneous function of degree one $H\colon\mathbb{R}^2 \rightarrow [0,+\infty[$. Under suitable assumptions on $\Omega$ and $H$, we also characterize the minimizers.

Keywords: Anisotropic perimeter, relative isoperimetric inequalities, Wulff shape.

MSC: 52A40

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