Bochner space

Abstract: We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Emery (via energy and $$\Gamma _2$$ -calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $$L^2$$ -Wasserstein distance.

Abstract: Given a finite-dimensional Banach space E and a Euclidean norm on E, we study relations between the norm and the Euclidean norm on subspaces of E of small codimension. Then for an operator taking values in a Hilbert space, we deduce an inequality for entropy numbers of the operator and its dual. In this note we study the following problem: given an n-dimensional Banach space E and a Euclidean norm 1j j 112 on E and 0 An such that (*) 11x12 < M*f(1 A)IIxlI for x E F. Here M* denotes the Levy mean of the dual norm of E (see the notation below). This problem was considered by V. Milman, who proved in [18] that estimate (*) holds for a certain exponential function f. The estimate was improved later in [10] to f(1 A) < K/(1 A), where K is a universal constant. The latter result turned out to be important for various applications (cf. [1, 15, 11, 19]). The main result of this note proves (*) with the function f(1 A) < K/v'?. This estimate, besides being optimal (up to a logarithmic factor), can be used to compare entropy numbers of an operator and its dual for operators taking values in a Hilbert space. Let us recall some notation. Let E be an n-dimensional Banach space; i.e., E = (Rn, * 11). Let [.,] be an inner product on Rn, and let j . j be the associated Euclidean norm on Rn defined by I11xI11 = [x,x]1/2, for x E Rn. Let BE be the closed unit ball in E. Set (1) gIIxII*=sup{I[x,y]IIyEBE} forxERn. Clearly, (Rn, jj *I1) can be identified with the dual space E*. Let S = {x E Rnl II xjII = 1}, and let ti be the normalized rotation invariant measure on S. Define the Levy means M and M* by M= (If I1xl2d}) , M,= (j !1XI2Idu) We shall employ a similar notation in a context of symmetric convex bodies. For a closed symmetric convex body V c Rn, by V* we denote the dual body defined byV* = {x Rnl [X,y]I < 1 forallyEV}. Received by the editors May 29, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B20; Secondary 47B10. lResearch partially supported by NSERC Grant A8854. (?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page