What is the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions, denoted by $d_\gh(B^m,B^n)$, for $m>n$? Note that the lower bound coming from the stability of persistent homology is zero, since all balls possess identical (trivial) persistent homology. To establish non-trivial lower bounds, we exploit the Borsuk--Ulam theorem. For any $m>n \ge 1$, we prove that $d_\gh(B^m,B^n)\ge \tfrac{1}{1+\alpha_n}>\frac{1}{2}$, where we determine $\alpha_n$ explicitly. We also prove that $d_\gh(B^m,B^n)\to 1$ as $m\to \infty$ and that $d_\gh(B^m,B^n)<1$ for all finite $m>n\geq 1$.
What becomes of Grothendieck's inequality when its Hilbert-space vectors are replaced by vectors in $\ell_p^r$ and $\ell_{p'}^r$? For an $m\times n$ matrix, we show that the universal dimensional loss is governed not by $r$ alone, but by the effective dimension \[ d=\min\{r,m,n\}. \] More precisely, the optimal constan...
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In parti...
For every finite family of Euclidean balls in $\mathbb{R}^n$, $n\ge2$, and every $0<\delta<1/2$, we select a subfamily whose $(1+\delta)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_n\delta^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \cite{HL}....
We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let $\psi:\N\to(0,\infty)$ be a non-increasing approximation function and let $\tau_\psi$ denote its lower order. For $m,n\ge1$ and $\boldsymbol\theta\in\mathbb R^m$, denote $\Exa...
We characterize Gromov-compactifiability of Wasserstein spaces over Polish metric spaces for $1\leq p\leq\infty$. The space $\mathcal{P}_1(X)$ is Gromov-compactifiable if and only if $X$ is bounded and Gromov-compactifiable, whereas $\mathcal{P}_p(X)$ is Gromov-compactifiable for every $1<p<\infty$. At the other endpoi...
Let $M$ be a closed connected Riemannian manifold and let $\mathcal P_s$ be the fractional perimeter on $M\times\mathbb R^k$ defined through its heat semigroup, where $k\ge1$ and $0<s<1$. We prove that the straight cylinder $M\times B_R$, with $B_R\subset\mathbb R^k$ a ball of radius $R$, uniquely minimizes $\mathcal P...
J. Seo· 0 citations
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