19 problems

Lipschitz magnitude conjecture. Let $T\subseteq L_1$ be tractable. Then

Magnitude conjecture. 1. We have

Let $X$ be a finite metric space with distance matrix $D$. Write $\operatorname{con}(D)$ for the concentration of a gauging of $D$, allowing the value $\infty$. A microscopic weigh…

Let $Y$ be a compact metric space of strictly negative type. Its magnitude function is denoted by $mu_Y$, and its maximal distance energy by $M(Y)$. Compact maximal-energy derivati…

Let $F$ be a skew finite subset of $\boldsymbol{\backslashell}_1^N$, and let $r\in(0,\operatorname{skew}(F)/2)$. Define … so that $k$ is the number of $1$-dimensional faces in…

Let $U$ be a finite-dimensional, positive definite normed space. For a finite subset $X\subseteq U$, write $|X|$ for its magnitude, and equip the set of finite subsets of $U$ with…

Let $E=(\mathbb{R}^n,\left\Vert\cdot\right\Vert)$ be a hypermetric normed space, and let $K\in\mathcal{K}^n$, the family of compact convex bodies in $\mathbb{R}^n$. Let…

Let $L_1$ denote the space of integrable functions, and let $K\subseteq L_1$ be compact and convex. Write $\mu_1^{L_1}(K)$ for the supremum of the first Holmes–Thompson intrinsic v…

Variant of the Leinster–Willerton conjecture. There are universal constants $\gamma_{0,n},\gamma_{1,n},\ldots,\gamma_{n,n}$ such that, for every such $X$,

Let $A$ be a compact $\ell_1$-convex set, meaning a geodesic subset of $\ell_1^n$, and let $V'_i(A)$ denote its $i$th $\ell_1$-intrinsic volume. These volumes are defined by the St…

Let $A$ be a compact convex subset of $\ell_2^n$, and let $V_i(A)$ denote its $i$th intrinsic volume. Magnitude conjecture. … This was previously conjectured by Leinster and Willer…

Let $K$ be a convex body in $\mathbb{R}^3$, and let $V_0(K)$ denote its zeroth intrinsic volume. Three-dimensional convex magnitude conjecture. The magnitude of $K$ should be … The…

Let $X\subseteq\mathbb{R}^2$ be a compact set suitable for the perimeter and classical Euler characteristic to be defined. Here $|tX|$ denotes the magnitude of the dilation of $X$…

Let $A$ be a compact $\ell_1$-convex subspace of $\ell_1^N$, and let $V'_i(A)$ denote its $i$-th $\ell_1$-intrinsic volume. The $\ell_1$ magnitude–intrinsic-volume conjecture. … Th…

Let $A$ be a compact convex subspace of $\ell_2^N$. For $0\leq i\leq N$, let $V_i(A)$ denote its $i$-th intrinsic volume, and let $\omega_i$ be the volume of the Euclidean unit bal…

Let $X$ be a space for which the magnitude $|X|$ and penguin valuation $P(X)$ are defined, and let $tX$ denote $X$ with all distances scaled by $t>0$. The penguin valuation is … wh…

Let $K$ be a convex subset of Euclidean space. Write $P(K)$ for its penguin valuation, … where $\mu_i(K)$ is the $i$th intrinsic volume and $\omega_i$ is the volume of the unit $i$…

Strong magnitude conjecture. There are unique functions $P,q\colon\mathcal{C}\to\mathbb R$ such that:

Weak magnitude conjecture. There is a unique function $P\colon\bar{\mathcal{C}}\to\mathbb R$ such that: