A vertex-skipping property for minimal perimeter sets in a convex container

Abstract

Given a open convex set K \subset \R^3 with a vertex-type singular point P \in \partial K, we focus on the behaviour around P of the boundary of a set E of minimal perimeter in K. The first part of the Seminar is devoted to prove that, provided K is a cone with a vertex at P, the reduced boundary of any perimeter minimizer in K skips P. This fact will be proved introducing a suitable notion of first variation on \partial E that does not send points of the closure of \partial E \cap \partial K onto points of \int K.
In the second part of the Seminar, I will expose a blow-up type argument in order to generalize this skipping-property to the case of general convex sets. 
These results are part of my PHD thesis under the supervision of Professor Gian Paolo Leonardi.

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