Let $S^n$ be the unit round sphere with its intrinsic angular metric, normalized so that $\operatorname{diam}S^n=\pi$. For finite homogeneous metric spaces $X$, put \[ \delta_n=\inf_X d_{GH}(X,S^n). \] The main open problem is whether $\inf_{n\ge2}\delta_n>0$. Gelander's theorem gives $\delta_n>0$ in each fixed dimension, but not uniformly. An abstract cross-polytope construction gives the universal upper bound $\delta_n\le\pi/4$. In the opposite direction, ChatGPT combines the passage from small Gromov--Hausdorff error to an approximate finite action on the sphere, logarithmic stability of approximate inner-product-preserving maps due to Cuesta, operator-norm stability of almost representations, and Green's width theorem for finite transitive sets. This gives the quantitative bound \[ \delta_n\ge \frac{c}{(1+\log(n+1))^2} \] for all sufficiently large $n$.
Let $X=G/K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=\Gamma\backslash X$. Sarnak's local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\|\phi\|_\infty\ll T^{(n-r)/2}$. We prove $o(T^{(n-r)/2})$ locally uniformly on e...
Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codime...
We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put $\zeta_n:=\arccos(-\tfrac{1}{n+1}),$ the common geodesic distance between distinct vertices of a regular simplex with $n+2$ vertices inscribed in $\mathbb{S}^n$. We prove that $$ d_{\mathrm{GH}}(\...
Donghan Kim, Sunhyuk Lim, Facundo Mémoli· 1 citation· ⚡1
We prove the Gromov--Ros conjecture for every rank-one symmetric space of noncompact type: finite-perimeter isoperimetric regions are precisely the geodesic balls, up to ambient isometry and null sets. The proof is organized uniformly in the root multiplicities $(p,q)$. We introduce the volume radius \[ R(r)^n=n\int_0^...
Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as...
Andrii Arman, A. Bondarenko, A. Prymak et al.· 0 citations
For every $N\ge 2$, we prove that the Bergman metric on the regular locus of a finite ball quotient $\mathbb{B}^N/\Gamma$, where $\Gamma\subset \mathrm{U}(N)$ is finite and fixed-point-free, is K\"ahler-Einstein if and only if $\Gamma$ is trivial. Consequently, if $\Omega$ is an $N$-dimensional normal Stein space with...
Soumya Ganguly, S. Sahi· 1 citation
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