Moment-entropy inequalities. (English) Zbl 1053.60004

It is shown that the product of the Rényi entropies \(N_{\lambda}(X)\) and \(N_{\lambda}(Y)\) of two independent random vectors \(X,Y\) in \(\mathbb{R}^n\) provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors. Namely, for real \(p\geq 1\) and real \(\lambda>n/(n+p)\) fixed, if \(X,Y\) have finite \(p\)th moment, then \(E(| X\cdot Y| ^p)\geq c_1[N_{\lambda}(X)N_{\lambda}(Y)]^{p/n}\), where the best possible \(c_1\) is explicitly given, and a condition for equality is provided. This very interesting result contains important geometry. For instance, if \(K,L\subset\mathbb{R}^n\) are compact, then it follows that \(\omega_n^2\max_{x\in K,y\in L}| x\cdot y| ^n\geq V(K)V(L)\), where \(\omega_n\) is the volume of the unit ball in \(\mathbb{R}^n\), and \(V\) is the Lebesgue measure. This inequality generalizes the well-known Blaschke-Santaló inequality, which is obtained when \(K\) is an origin-symmetric convex body and \(L\) is its polar body.
The authors first establish connections and inequalities between Rényi entropy and dual mixed volumes of random vectors. Moreover, a star body \(S_pX\) associated to a random vector \(X\) is defined. Then, for \(p\geq 1\), an equality is shown between \(E(| X\cdot Y| ^p)\) and the dual volume of \(X\) (respectively, \(Y\)) and the polar \(L_p\)-centroid body \(\Gamma^{\ast}S_pY\) (respectively, \(\Gamma^{\ast}S_pX\)). By employing these lemmas and some further arguments, the main result is derived


60D05 Geometric probability and stochastic geometry
52A22 Random convex sets and integral geometry (aspects of convex geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
60E15 Inequalities; stochastic orderings
94A17 Measures of information, entropy
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