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We establish a sharp affine $$L^p$$ Sobolev trace inequality by using the $$L_p$$ Busemann-Petty centroid inequality. For $$p = 2$$ , our affine version is stronger than the famous sharp $$L^2$$ Sobolev trace inequality proved independently by Escobar and Beckner. Our approach allows also to characterize all extremizers in this case. For this new inequality, no Euclidean geometric structure is needed. [ABSTRACT FROM AUTHOR]

We show that the elastic energy $$E(\gamma )$$ of a closed curve $$\gamma $$ has a minimizer among all plane simple regular closed curves of given enclosed area $$A(\gamma )$$ , and that the minimum is attained for a circle. The proof is of a geometric nature and deforms parts of $$\gamma $$ in a finite number of steps to construct some related convex sets with smaller energy. [ABSTRACT FROM AUTHOR]