We prove that any $n$-dimensional normed space is at Banach--Mazur distance $O(n^{2/3})$ from $\ell_\infty^n$.
In this paper, we show that the Ces\`{a}ro operator on the Hardy space $H^p$, $0<p<\infty$, is cyclic. Our techniques will involve semigroups.
We prove stability estimates for the volume of central hyperplane projections of unit balls in $\ell_p^n(\mathbb{R})$, establishing the dual counterpart of the result of Chasapis, Nayar and Tkocz on stability of sections.
We prove that $M_n(\mathbb{C})$ contains no abelian Kadison--Singer algebra for any $n \geq 3$.
Let $f:\mathbf P^n_{\mathbb C}\to\mathbf P^n_{\mathbb C}$ be an endomorphism of degree greater than one. We prove that every irreducible Zariski closed subset $V$ with $f^{-1}(V)=V$ is a linear subspace.
We prove an exact isoperimetric inequality for cubes and tori under the $\ell_{\infty} metric. As a corollary, we recover the celebrated grid edge-isoperimetric inequality of Bollob\'{a}s-Leader.
We prove that if $E$ is a UMD symmetric Banach function space on $(0,\infty)$, then $E(\mathcal{M},\tau)$ is UMD for every semifinite von Neumann algebra $\mathcal{M}$ equipped with a faithful normal semifinite trace $\tau$. This resolves in the affirmative an open problem that has circulated in the non-commutative wor...
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