Fourier analytic methods in convex geometry

    The following definitions and Theorem were presented by Schneider in his talk.

Let ${\mathcal K}^n$ denote the class of convex bodies in $\mathbb{R}^n$. A Minkowski class $\mathcal{M}$ is a subset of ${\mathcal K}^n$ that is closed in the Hausdorff metric, under Minkowski linear combinations and translations. A body $K$ is a generalized $\mathcal{M}$-body if there are two bodies $M_1,M_2 \in \mathcal{M}$ such that $K+M_1=M_2$.

Let $G$ be a subgroup of $GL(n)$. We say that $\mathcal{M}$ is $G$-invariant if whenever $K \in M$ and $g \in G$, we have $gK \in \mathcal{M}$. Given $B \in {\mathcal K}^n$, $G \subset GL(n)$, we define $M_{B,G}$ as the smallest Minkowski class containing $B$ that is $G$-invariant.

Theorem (Schneider, F. Schuster) Let $B\in {\mathcal K}^n$ be non-symmetric. Every neighborhood of $B$ contains an affine image $B'$ of $B$ such that the generalized $M_{B',SO(n)}$ bodies are dense in ${\mathcal K}^n$.

    Let $K$ be a symmetric convex body. We define the zonoid ratio as \[ zr(K)=\min_{Z \in {\mathcal Z}, Z \subset K} \frac{|K|^{1/n}}{|Z|^{1/n}}. \] The volume ratio is defined as \[ vr(K)=\min_{E \, ellipsoid, \, E \subset K} \frac{|K|^{1/n}}{|E|^{1/n}}. \] Clearly, $vr(K) \geq zr(K)$. Y. Gordon and M. Junge proved that there is a constant $C_0>0$ such that every centrally symmetric convex body $K$ in $R^n$ has a section $L$ of dimension $[n/2]$ such that $zr(L)\geq C_0 \, vr(K)$.

Since by K. Ball the local unconditional constant $\chi_u(L)$ is $\ge zr(L)$ for every convex $L$, and these parameters of the local theory are always hard to compute, it follows that $C_0\, vr(K)$ yields an easy computational lower bound for some $[n/2]$ dimensional section $L$ of $K$.

    Which projection or/and section data are needed to determine non-symmetric convex bodies? Several known cases: A non-symmetric convex body is uniquely determined if we know:

• The volume of sections passing through a given interior point and the centers of gravity of those sections.
• The mean width and the Steiner points.
• The brightness function and the illumination function.

    Given $t \leq 2\sqrt{2}-2$, the minimal slab is in the direction of $(1,0,\ldots,0)$.

    We say that a measurable function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a ridge function if it is constant in one direction, i.e., $f(x)=g(\langle x, v \rangle)$ for some $g:\mathbb{R} \rightarrow \mathbb{R}$, and $v \in S^1$. Given $n$ ridge functions $f_1,\ldots,f_n$ with directions $v_1,\ldots,v_n$ respectively, the set $E=\{x \in \mathbb{R}^2: (f_1+\cdots,f_n)(x)\geq 0 \}$ is uniquely determined among all measurable sets by the X-rays in the directions of $v_1,\ldots,v_n$.

It is known that there are sets of four directions such that every planar convex body can be determined by X-rays in those directions (for example, the directions of $v_1=(1,0)$, $v_2=(0,1)$, $v_3=(2,1)$ $v_4=(-1,2)$ work). However, no set of three directions is enough.

    For $t>0$, $a \in S^{n-1}$, consider the slab \[ Sl(a,t)=\{x \in \mathbb{R}^n : \, \|x\|_\infty \leq 1, \, |\langle x,a \rangle|\leq t \} \] and the section \[ S(a,t)=\{x \in \mathbb{R}^n : \, \|x\|_\infty \leq 1, \, \langle x,a \rangle = t \}. \] Denote $V(a,t)=vol_n(Sl(a,t))$ and $A(a,t)=vol_{n-1}(S(a,t))$.

Let $f_m=(1,\ldots,1,0,\ldots,0)/\sqrt{m} \in S^{n-1}$, where $1 \leq m \leq n$ and there are exactly $m$ ones in $f_m$. It is known (Hensley, Ball;Oleskiewicz-Pelcynski) that $1 \leq A(a,0) \leq \sqrt{2}$, with equality in the left attained for $a=f_1$, and on the right for $a=f_2$. (In the complex case, the result is $1 \leq A(a,0) \leq 2$).