14 problems

Let $K$ be a convex body in a normed space, let $x\in\operatorname{int}K$, and let $B_n(K,x)$ denote the Bernstein factor at $x$ for polynomials of degree at most $n$. Write $w(K)$…

Slicing conjecture. The slicing constant is universally bounded:

Let $M,K$ be convex bodies in $\mathbb{R}^n$ with $M\subset\operatorname{int}K$ and $n\geq 3$. For $x\in\operatorname{bd}K$, let the graze $\Sigma(M,x)$ be the set of contact point…

Macbeath point conjecture. The inclusion

Let $L,K\subset \mathbb{R}^{n}$ be convex bodies, with $n\geq 3$, and suppose that $L\subset \operatorname{int}K$. For $x\in\mathbb{R}^{n}\setminus K$, let $S(L,x)$ denote the boun…

For a convex body $K\subseteq\mathbb{R}^n$, define … where $K^\circ$ is the polar body. Ball proved that $B(K)\leq B(B_2^n)$ for unconditional convex bodies. Ball's conjecture. The…

Let $K\subset\mathbb{R}^d$ be a symmetric convex body, let $X$ be uniformly distributed on $K$, and let $\Sigma$ be the covariance matrix of $X$, with entries … Write $\|\Sigma\|$…

Let $\phi$ be the norm defining the anisotropic curvature measures $\mathcal{C}_m^\phi(C,\cdot)$ of a convex body $C\subseteq\mathbf{R}^{n+1}$. Anisotropic curvature-measure charac…

Let $\phi$ be a smooth, strictly convex norm on $\mathbf{R}^{n+1}$, with Wulff shape $\mathcal{W}^\phi=\{x\in\mathbf{R}^{n+1}:\phi^*(x)=1\}$. Let $X_0:M\to\mathbf{R}^{n+1}$ be a cl…

Let $K\subset\mathbb{R}^n$ be an $o$-symmetric convex body, and let $\Pi K$ be its projection body, defined by … Its polar projection body is $\Pi^\circ K=(\Pi K)^\circ$. Petty's c…

Let $d\geqslant 2$ be fixed. For a $d$-dimensional convex body $C$, let $\delta(C)$ and $\theta(C)$ denote its packing and covering densities by congruent copies. Kuperberg's conje…

Let $K\subset \mathbb{R}^{2n}$ be a convex body covered by a finite set of convex bodies $\{K_i\}$. Subadditivity conjecture. One has … This is a special case of the subadditivity…

Generalized lattice-point bound conjecture. Under these conditions,

Let ${\bf K}$ be an arbitrary $0$-symmetric convex body in ${\mathbb R}^3$. A truncated octahedron of the form ${\bf T}-{\bf T}$ is obtained from an arbitrary tetrahedron ${\bf T}$…