Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality (Q418724)

A functional version of the affine isoperimetric inequality is obtained for \(s\)-concave functions and for their natural functional extension, the log-concave functions on \(\mathbb{R}^n\). This inequality may be interpreted as an inverse form of a logarithmic Sobolev inequality for entropy. As a consequence, an inequality inverse to the Poincaré inequality for the Gaussian measure is derived by linearization. Meanwhile, if \(K\subset \mathbb{R}^n\) is a convex body with a smooth boundary \(\partial K\), useful evaluations of the generalized Gaussian curvature and the outer unit normal vector to \(\partial K\) at \(x\in \partial K\) are obtained. This allow the authors to prove a precise formula to compute the affine surface area for an \(s\)-concave function \(f\) on \(\mathbb{R}^n\), in terms of derivatives of \(f\), by means of the convex body \(K_s (f)\) associated to \(f\) by \(K_s (f):=\{(x,y)\in \mathbb{R}^n \times \mathbb{R}^n : x\in \operatorname{supp}(f), \|y\|\leq f^{\frac{1}{s}}(x)\}\).