Let and be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space , where the Young function behaves like , , for large, into the Zygmund space . We also study the same problem for the embedding of the generalized Lorentz-Sobolev space , , , , embedded into the Zygmund space .
We derive the equivalence of different forms of Gaussian type shift inequalities. This completes previous results by Bobkov. Our argument strongly relies on the Gaussian model for which we give a geometric approach in terms of norms of barycentres. Similar inequalities hold in the discrete setting; they improve the known results on the so-called isodiametral problem for the discrete cube. The study of norms of barycentres for subsets of convex bodies completes the exposition.
We define a Sobolev space by means of a generalized Poincaré inequality and relate it to a corresponding space based on upper gradients.