
## 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.

#### 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.

#### 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?
• #### 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.
• #### 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.

#### Problem 2.5.

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