We study the $L^p(\mu)$-norm of the metric projection onto a closed, convex set $C \subset \mathbf{R}^n$ when $\mu$ is the uniform measure on the sphere or the standard Gaussian measure on $\mathbf{R}^n$. Up to universal constants, we determine the optimal reverse H\"older inequalities (i.e., $L^q-L^p$ estimates for $q>p$) for both settings and for all $1 \leq p<q \leq \infty$. The optimal constants in these inequalities depend polynomially on the dimension $n$. We establish upper bounds for the expected norm of the metric projection for a wide class of probability measures. Our inequalities improve and extend previous results of S. Chatterjee.
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 codimension-one refinement of the known almost-everywhere partial regularity, and the exponent is sharp. More generally, for a convex viscosity subsolution of $\sigma_k(D^2u)\ge\lambda>0$, we obtain Hausdorff bounds for strata defined by the affine dimension of all supporting contact sets. The proof combines a support-dependent Chou--Wang barrier argument, an estimate for the product of the smallest $k$ semiaxes of a John ellipsoid, and Mooney's convex section-covering theorem. As a direct analytical consequence, the full distributional Hessian is absolutely continuous and $u\in W^{2,1}_{\mathrm{loc}}(\Omega)$, yielding a $k$-Hessian counterpart of the $W^{2,1}$ regularity known for singular Monge--Amp\`ere solutions. In a logically separate structural part, we characterize the distinguished number of flat directions, $n-k+1$, by an asymptotic infimum mean-value formula over affine sections, and explain how this mean-value heuristic leads to the supporting-contact geometry used in the proof.
Let $K\subset\R^n$ be a convex body in isotropic position. We prove the optimal mean-width estimate \[ M^*(K)\leq C\sqrt{n\log n}. \] The main new ingredient is a geometric inequality relating the Gaussian mean of the support function to its mean under the uniform measure on $K$, obtained through a heat-flow argument. Combined with the Gaussian-log-concave comparison of Eldan and Lehec and the newly available dimension-free bound on the third-moment parameter $\kappa_n$, this yields the result. The boundedness of $\kappa_n$ also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields \[ M(K)M^*(K)\leq C\log n, \] extending Pisier's $MM^*$ estimate to non-symmetric convex bodies and to the isotropic position.
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 every quotient. On finite-volume real hyperbolic manifolds this is uniform in the expanding cusp range $y\leq T^\beta$, $\beta<1/2$. If the injectivity radius is bounded below, we prove the global estimate $T^{(n-r)/2}(\log T)^{-r/2}$.
We prove a sufficient criterion for closed subspaces of operator spaces containing the finite-rank operators to have the uniform ball-covering property. Let $F$ be a separable uniformly convex Banach space, and let $\Lambda_F>1$ be a constant determined by its modulus of convexity. If $F$ has the $\pi_\lambda$-property for some $1\leq \lambda<\Lambda_F$, then for every Banach space $E$ with separable dual, every closed subspace of $\mathcal{B}(E,F)$ containing $\mathcal{F}(E,F)$ has the UBCP. The proof uses a contraction estimate for near-metric finite-rank projections on uniformly convex spaces. We use this estimate to construct uniform ball coverings for the corresponding operator spaces. As applications, we obtain the UBCP for closed operator subspaces whose range spaces are vector-valued $L_p$-spaces, or separable uniformly convex $\mathcal{L}_{p,C+}$-spaces.
Fix an integer $p\geq 2$ and $0<s<1$. We construct an ergodic $\times p$-invariant measure $\mu$ on $\mathbb T^2$ of dimension $s$, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass $O(w^s)$ to every planar tube of width $w$. The result is sharp at the endpoint $s=1$ as shown by Py\"or\"al\"a, Shmerkin, Suomala and Wu (2025). In contrast, we show that every quasi-Bernoulli $T_p$-invariant measure with positive dimension admits a dimension-dropping projection.
Amir Algom· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
