*Published Paper*

**Inserted:** 11 nov 2018

**Last Updated:** 11 nov 2018

**Journal:** Ann. Inst. H. PoincarĂ© - Anal. Non LinĂ©aire

**Volume:** 25

**Number:** 4

**Pages:** 803-832

**Year:** 2008

**Abstract:**

A set is called "calibrable" if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the $\phi$-calibrability of bounded convex sets in $\mathbb R^N$, with respect to a norm $\phi$ (called anisotropy), by the anisotropic mean $\phi$-curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex body $C$ satisfying a $\phi$-ball condition contains a convex $\phi$-calibrable set $K$ such that, for any $V\in [

K

,

C

]$, the subset of $C$ of volume $V$ which minimizes the $\phi$-perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data by the characteristic function of a bounded convex set.

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