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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

MSC:

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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