Functional affine-isoperimetry and an inverse logarithmic Sobolev   inequality
  We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincar'e inequality for the Gaussian measure. 
S. Artstein-Avidan, B. Klartag, C. Schuett and E. Werner
math.FA