21 problems

Let $A$ be a compact set in $d^d$ for some $d\in\mathbb{N}$. For $k\in\mathbb{N}$, write … The volume is the Lebesgue measure, denoted by $\operatorname{vol}$. Bobkov–Madiman–Wang…

Altmann's conjecture. For every Minkowski decomposition of $Q$ into lattice polytopes, project the summands into $T^1_Y$ and take their linear hull. The union of all linear subspac…

Let $Q$ be an MV polytope of type $A$, and let $Q(\theta_I)$ denote the Schubert matroid polytope associated with $I\subseteq [n]$. For collections…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$. For a subset $A\subseteq\mathbb{R}^n$, define its $k$-fold Minkowski sum by … Call $A$ balanced if $a\in A$ and…

Unsupported-vector conjecture. Since method $\texttt{\textsc{u}}$ generates local sets with many unsupported vectors, the relative minimum generator-set cardinality should be lower…

For convex bodies $A,B_1,B_2\subset\mathbb{R}^n$, let $c_{n,2}$ be the smallest constant such that … The previously established values include $c_{2,2}=1$ and $c_{3,2}=4/3$. Fradel…

Let $P\subset\mathbb{R}^d$ be a flexible polytope, meaning that it admits a continuous deformation preserving its combinatorial type and edge lengths that is not a continuous isome…

Let $n\geq1$ and let $A,B,C$ be zonoids, hence convex bodies in $\mathbb{R}^n$. Zonoid supermodularity conjecture. … This is motivated by the proved case in which the first summand…

Let $n\geq 2$. For a convex body $A$ and compact sets $B,C\subset\mathbb{R}^n$, consider their Minkowski sums. Supermodularity conjecture. … This extends the known one-dimensional…

Let $d$ be a positive integer, let $m$ be a real number, and let $K_1,\ldots,K_d$ be convex bodies in $\mathds{R}^d$ satisfying … For an integer $\ell$ with $1\leq\ell\leq d$, the…

Let $k\geq 2$, and let $A_1,A_2,\ldots,A_{k+1}$ be compact sets in $\mathbb{R}^d$. For a finite family of sets, $\sum_i A_i$ denotes their Minkowski sum, and $\operatorname{vol}$ d…

For integers $k,n$ with $k>3$, let $\mathcal{A}_{k,n}$ denote the Minkowski sum of $n$ copies of the moment-curve segment $\{(t,t^2,\ldots,t^k)\mid -1\leq t\leq 1\}$. Unique bounda…

For integers $k,n$ with $k>3$, let $\mathcal{A}_{k,n}$ denote the Minkowski sum of $n$ copies of the moment-curve segment $\{(t,t^2,\ldots,t^k)\mid -1\leq t\leq 1\}$. Let…

Let $\mathcal{K}^n$ denote the family of convex bodies in $\mathbb{R}^n$, let $K_i\in\mathcal{K}^n$ for $i\in[j]$, and let…

Reducibility conjecture. A $d$-irreducible tiling of the $d$-dimensional space has a reducible parallelotope $P=P_1\oplus P_2$.

Let $A_1,\ldots,A_k$ be measurable subsets of Euclidean space, and interpret volume as the order-zero analogue of Rényi entropy. BMW's conjecture. Analogues for volumes of fraction…

Let $A$ and $B$ be convex bodies in $\mathbb{R}^n$. For every vector $t\in\mathbb R^n$, consider the translate $A+t$ and its difference from $B$. Minkowski summand conjecture. If…

Let $A$ and $B$ be convex bodies in $\mathbb R^n$. The set $(A+B)\setminus A$ is the difference between a Minkowski sum and one summand. Homological sphere conjecture. The set…

Let $\mathcal{K}^n$ be the space of convex bodies in $\mathbb{R}^n$. A Blaschke-Minkowski homomorphism is a map from convex bodies, equipped with Blaschke addition, to convex bodie…

Let $n$ be the dimension, and let $\mathcal{K}^n$ denote the space of convex bodies in $\mathbb{R}^n$. A Minkowski endomorphism is a continuous, rotation-intertwining, Minkowski-ad…

Minkowski equality conjecture. Equality holds if and only if $K_0$ and $K_1$ are homothetic. The conjecture concerns the equality case in the Minkowski inclusion for exterior dista…