A stability estimate for the Aleksandrov-Fenchel inequality under regularity assumptions
Since the family of all convex bodies in \({\mathbb R}^{n+1}\), equipped with the Minkowski addition and multiplication by non-negative real numbers, does not constitute a vector space, the author considers the vector space \({\mathcal H}^{n+1}\) of formal differences of convex bodies: thus, \textit{hedgehogs} are just the geometrical realizations of elements of \({\mathcal H}^{n+1}\) in \({\mathbb R}^{n+1}\). Then, mixed volumes of convex bodies are extended to symmetric (\(n+1\))-linear forms on \({\mathcal H}^{n+1}\), and certain parts of the classical Brunn-Minkowski theory are also extended to \({\mathcal H}^{n+1}\). In the paper under review, the author obtains a strengthening of the famous Aleksandrov-Fenchel inequality for convex bodies in the form of a stability estimate, working under suitable regularity assumptions and in the setting of hedgehogs. More precisely: identifying convex bodies (and hedgehogs) with their respective support functions, he writes \(v(h,k;l):=v(h,k,l_3,\dots,l_{n+1})\) to denote the mixed volume of the sets \(H,K,L_3,\dots,L_{n+1}\) whose support functions are \(h,k,l_3,\dots,l_{n+1}\); then he shows that if \(K,L_i\), \(i=3,\dots,n+1\), are convex bodies of class \(C^2_+\) and \(H\) is a convex body of class \(C^2\), \[ v(h,k;l)^2-v(h,h;l)v(k,k;l)\geq\dfrac{v(k,k;l)^2}{4}\bigl(M_{(h,k;l)}-m_{(h,k;l)}\bigr)^2, \] where \(M_{(h,k;l)}\) and \(m_{(h,k;l)}\) are, respectively, the maximum and the minimum on \(\nu\in{\mathbb S}^n\) of the function \(v(h,\sigma_{\nu};l)/v(k,\sigma_{\nu};l)\), and \(\sigma_{\nu}\) is the support function of the unit segment in the direction \(\nu\) centered at the origin.
scientific article; zbMATH DE number 6677797