Digital Object Identifier (DOI) 10.1007/s00205-005-0387-0 Arch. Rational Mech. Anal. 179 (2005) 109–152 Crystalline Mean Curvature Flow of Convex Sets Giovanni Bellettini, Vicent Caselles, Antonin Chambolle & Matteo Novaga Communicated by L.C. Evans Abstract We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in IR N . This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat φ-curva- ture flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat φ-curvature flow starting from a compact convex set is unique. 1. Introduction In this paper we deal with the anisotropic mean curvature motion, which is defined as the gradient flow of the surface energy functional P φ defined as P φ (E) :=  ∂E φ ◦ (ν E )d H N -1 , E ⊂ IR N , where ν E is the outward unit normal to the boundary ∂E of E and φ ◦ (the surface tension) is a positively one-homogeneous and even function such that {φ ◦ ≦ 1} is a compact convex set with nonempty interior. We are particularly interested in the case when N ≧ 3 and {φ ◦ ≦ 1} is not smooth; in this respect, we say that the anisotropy φ ◦ is crystalline if {φ ◦ ≦ 1} is a polyhedron. Anisotropic mean curvature flow and its generalizations are used to describe several phenomena in material science and crystal growth, see for instance [18, 38, 32]. From the mathematical point of view, the analysis was initiated by J. Taylor [38, 39] and developed further in [2, 17, 1, 31, 41]. In comparison with more famil- iar geometric evolutions such as mean curvature flow (corresponding to the choice