Following the equivalence between logarithmic Sobolev inequalities and hype
rcontractivity showed by L. Gross, we prove that logarithmic Sobolev inequa
lities are related similarly to hypercontractivity of solutions of Hamilton
-Jacobi equations. By the infimum-convolution description of the Hamilton-J
acobi solutions, this approach provides a clear view of the connection betw
een logarithmic Sobolev inequalities and transportation cost inequalities i
nvestigated recently by F. Otto and C. Villani. In particular, we recover i
n this way transportation from Brunn-Minkowski inequalities and for the exp
onential measure. (C) 2001 Editions scientifiques et medicales Elsevier SAS
.