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We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convexcones, aswell as a weighted sharp trace Sobolev inequality on epigraphs ofconvex functions. This is done by using a generalized Borell-Brascamp-Liebinequality, coming from the Brunn-Minkowski theory.

Sharp trace Gagliardo-Nirenberg-Sobolev inequalities for convex cones, and convex domains