In this paper, we study the symmetrized Talagrand inequality that was proved by Fathi and has a connection with Blaschke-Santaló in convex geometry. As corollaries of our results, have several refined functional inequalities under some conditions. We also give an alternative proof Fathi's on real line.

Quantum Euclidean spaces, as Moyal deformations of are the model examples noncompact noncommutative manifold. In this paper, we study quantum space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example semifinite spectral triple nonflat geometric structure. We develop abstract symbol calculus for pseudo-differential operators noncommuting deriv...

The purpose of this paper is to give an elementary proof of the fact that a Banach space in which there exist projection transformations of norm one on every two-dimensional linear subspace is a euclidean space. S. Kakutani [ l ] has pointed out that a modification of a proof due to Blaschke [2] will prove this theorem. F. Bohnenblust has been able to establish this theorem for the complex case...