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2. Symmetry and convexity

    1. Polynomially integrable bodies

          Let $A_{K,u}(t) = |K \cap \{ tu + u^\perp \}|_{n-1}$ be a parallel section function of $K$, then an infinitely smooth convex body $K$ in $\R^n$ is called polynomially integrable if its parallel section functions $A_{K,u}(t)$ are polynomials of $t$ (on its support) for all $u \in S^{n-1}$.

      It is known that the only smooth bodies in odd dimensions are ellipsoids, and in even dimensions, such bodies do not exist.

      Add remark

      Problem 2.1.

      Replacing volume of the sections by the surface area (or any intrinsic volume), is it true that the only bodies in even dimensions with such property are ellipsoids?

        • Question by Makai and Martini

              It is known that if for any direction $u \in S^{n-1}$, a parallel section function $A_{K,u}(t)$ has maximum at $t = 0$, then body $K$ is symmetric.

          Add remark

          Problem 2.2.

          What if we pose the same question but replacing volumes by surface areas, that is $\widetilde{A}_{K,u}(t) = |\partial(K \cap \{tu + u^\perp\})|_{n-1}$, is body $K$ still symmetric?


            • Add remark

              Problem 2.3.

              Let $p > 1$ and $H \subset \R^n$ of dimension $k$. Consider a map $$ f(t_1, \dots, t_n) = \left| \begin{pmatrix} e^{t_1} & 0 & \cdots & 0 \\ 0 & e^{t_2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & e^{t_n} \end{pmatrix} B_p^n \cap H \right|_k, $$ where $B_p^n = \{ x\in \R^n: \ \sqrt[p]{|x_1|^p + \cdots + |x_n|^p} \leq 1\}$.

              Is $f$ log-concave jointly in $t_1, \dots, t_n$?
                  For cases $p = 1, 2$, the answer is given by Tkocz, Nayar. Note that when $p = \infty$, the statement is equivalent to log-Brunn-Minkowski.


                • Add remark

                  Problem 2.4.

                  Take $a_1, \dots, a_n \in \R$ and $v_1, \dots, v_n \in \R^n$ and define $$ f(t) = \int\limits_{\R^k} e^{-\max \limits_{ 1 \leq i \leq n } e^{ a_i t }|\left< x , v_i \right> |} \, dx. $$ Then $f(t)$ is log-concave is equivalent to log-Brunn-Minkowski. What is possible $f$ that we can use for Brunn-Minkowski?

                    •     
                      Theorem.
                      [Caffarelli] Let $T = \nabla \Phi: \mu \to \nu$ be a Brenier map, where $\mu$ and $\nu$ are probability measures with densities $e^{-V(x)}$ and $e^{-W(x)}$. Suppose that $$ \nabla^2 W(x) \geq K \, Id, $$ for some constant $K$. Then for any $e \in S^{n-1}$ $$ \sup\limits_{ x \in \R^n } \Phi^2_{ e e } \leq \frac1K \sup\limits_{x \in \R^n} V_{ee}, $$ where $\Phi_{e}$ denotes directional derivative.

                      Add remark

                      Problem 2.5.

                      If we replace condition in the theorem with $\nabla^2 W(x) \geq F(x) \, Id$, can we obtain better results?


                        • Add remark

                          Problem 2.6.

                          For a smooth convex domain $K$ in $\R^n$, Laplace equation with Dirichlet boundary conditions is $$ \begin{cases} \Delta u = \lambda u \ &\text{ on } K \\ u \equiv 0 &\text{ on } \partial K \end{cases} $$ Choose $u_1 > 0$ in $\text{int}\, K$, then $$ \text{Hess} \, \log{u_1} \leq 0? $$

                          True for $K = B_2$. Question is whether it is true for complex projective space $\mathbb{C}\mathbb{P}^n$ or for other symmetric K’s.

                              Cite this as: AimPL: Symmetry and convexity in geometric inequalities, available at http://aimpl.org/symconvgeomineq.