## 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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