We show that, for every pair of convex bodies $K,L\subset\mathbb R^n$, $$ \operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}. $$ The main point is to place $K$ and $L^\circ$ in isotropic position. We then consider a random orthogonal image of $L$ and control the corresponding operator norm by combining the isotropic mean-gauge estimate of Bizeul and Klartag with Letwin's recent dimension-free bound for the third-moment parameter appearing in their estimate. Our result improves the bound $ \operatorname{vr}(K,L)\leq C\sqrt n \log(n+1)$ proved by Giannopoulos and Hartzoulaki, which had remained the best general estimate for nearly two and a half decades.
We prove that, for every integer $t\geq 1$, the unit two-sphere admits a spherical $t$-design consisting of exactly $54t^2$ points. More generally, such a design exists with exactly $6q^2$ points for every integer $q\geq 3t$. The proof builds on the topological degree method of Bondarenko, Radchenko and Viazovska, usin...
Let $M(N)$ be the number of similarity classes of positive rational perfect cuboids with ordered edges $a,b,c$, space diagonal $g$, and squarefree invariant $N=\operatorname{sf}(abcg/2)$. We prove that $M(N)$ is finite for every positive squarefree $N$, without any restriction on the rank of the associated congruent-nu...
Let $K$ be an origin-symmetric full-dimensional convex polytope in $\mathbb R^n$, $n\ge 3$. We show that, for $-n+3<p<1$, the normed space $(\mathbb R^n,\|\cdot\|_K)$ embeds in $L_p$ if and only if it embeds in $L_1$. The case of $p\le 0$ is understood in the generalized sense. In particular, an origin-symmetric full-d...
Let $X_N=\{x_1,\dots,x_N\}\subset \Sph^2$ be a spherical $t$-design of strength $t\ge2$, and let $\mu_X$ be its empirical measure. Minkowski's theorem associates with $X_N$ a convex polytope $P_X\subset\R^3$, unique up to translation, whose facet normals are the design nodes and whose facets all have area $4\pi/N$; equ...
Let ${R^\ast}(n)$ be the least radius of a central circle to which nonoverlapping circles of radii $1,\ldots,n$ are externally tangent. We prove that ${R^\ast}(n)={C_\ast} n^2+o(n^2)$ and characterize ${C_\ast}$ by finite linear programs with an explicit error tending to zero. The reduction preserves arbitrary orders a...
We prove the lower bounds \[ B_\infty>0.51,\qquad L>0.51, \] where $B_\infty$ is the locally univalent Bloch constant and $L$ is Landau's constant, improving the bound $\frac{1}{2}+2\cdot10^{-8}$ of Chen and Shiba. The argument passes to the logarithm $g=\log f'$ of a normalized locally univalent Bloch function, where...
Frank Wikström· 0 citations
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