Santaló region of a log-concave function (Q2841075)

The paper is concerned with the Santaló region and the Floating body of a~log-concave function on \(\mathbb R^n\). The main result asserts that any relation involving Floating body and Santaló region of a~convex body is translated to a~relation of the Floating body and the Santaló region of an even log-concave function. More precisely, given a~convex body \(K\) in~\(\mathbb R^n\), the Santaló region with constant~\(t\) is defined as \(S(K,t)=\{x\in X\mid \text{vol} (K) \text{vol} (K^x)\leqslant t \text{vol}(B^n_2)^2\}\); here \(K^x\) is the polar body of~\(K\) with respect to~\(x\), \(B^2_n\) is the unit ball of~\(\mathbb R^n\) in the \(\ell^2\)-norm.NEWLINENEWLINENEWLINEThe main result is as follows. Let \(0<\lambda <1/2\) and \(0<d\) such that \(F(K,\lambda)\subset S(K,d)\) for every centrally symmetric convex body~\(K\). Then \(F(f,\lambda)\subset S(f,d)\) for every even log-concave function \(f:\mathbb R^n\to [0,\infty)\). Here, \(F(f,\cdot)\) and \(S(f,\cdot)\) are, respectively, the Santaló region and the Floating body of a~log-concave function~\(f\), and \(F(K,\lambda)\) is the Floating body of~\(K\).