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3. Geometric Inequalities


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      Problem 3.1.

      [Galyna Livshyts] Let $K, L \subset \mathbb{R}^n$ be convex bodies.

      Question 1: If $\gamma$ is a standard Gaussian measure and $0 \leq \lambda \leq 1$, is it true that \[\left[\gamma(\lambda K + (1 - \lambda) L)\right]^{1/n} \geq \lambda \gamma(K)^{1/n} + (1 - \lambda) \gamma(L)^{1/n},\] if $K$ and $L$ are origin-symmetric, i.e. $x \in K \implies -x \in K$, or are Gaussian barycentered, i.e. $\int_K \underline{x} \exp\left(-\frac{|\underline{x}|^2}{2}\right) d \underline{x} = 0$? The above is known to be true if $n=2$ or if the problem has many symmetries. Often this problem is referred to as the “Gardner-Zvavitch Conjecture".

      Question 2: This open problem was noted by Böröczky, Lutwak, Yang, and Zhang. Let $K$ and $L$ be origin-symmetric. We define the geometric average as \[K^{\lambda} L^{1-\lambda} \triangleq \left\{ \underline{u} \in \mathbb{R}^n : \langle \underline{x}, \underline{u}\rangle \leq h_K(\underline{u})^{\lambda} h_L(\underline{u})^{1 - \lambda}, \forall \underline{x} \in \mathbb{R}^n, ||\underline{x}||^2 = 1, h_K(\cdot) \text{ is the support function}\right\}.\] The conjecture is \[Leb\left(K^{\lambda}L^{1 - \lambda}\right) \geq (Leb(K))^{\lambda} (Leb(L))^{1-\lambda}.\] Note that there exists a counterexample if $K$ and $L$ are not origin-symmetric and it is know that the result of Question 2 implies the result of Question 1.

      Question 3: What is the random variable manifestation of Question 1?

      Question 4: Let $\gamma$ be a centered, Gaussian measure on $\mathbb{R}^n$ and $K$ an origin-symmetric body in $\mathbb{R}^n$. Consider \[f(t_1, t_2, \ldots, t_n) = \gamma(K[diag(e^{t_1}, e^{t_2}, \ldots, e^{t_n})])\] where for a vector $v \in \mathbb{R}^n$, $diag(v)$ is the $n \times n$ matrix with $v$ along the diagonal. Then is $f(\cdot)$ log-concave? If $\gamma$ is standard Gaussian, the result is true (Cordero, Maurey, Fradelize). It is know that the result of Question 4 implies the result of Question 2.

          Cite this as: AimPL: Entropy power inequalities, available at http://aimpl.org/entropypower.