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Consider an $n\times n$ matrix polynomial $P(\lambda)$ and a set $\Sigma$ consisting of $k \le n$ distinct complex numbers. In this paper, a (weighted) spectral norm distance from $P(\lambda)$ to the matrix polynomials whose spectra include…
Existing guarantees for algorithms sampling from nonlogconcave measures on $\mathbb{R}^d$ are generally inexplicit or unscalable. Even for the class of measures with logdensities that have bounded Hessians and are strongly concave outside a…
We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a…
Let $E\subset\rr$ be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$…
While there is extensive literature on approximation, deterministic as well as random, of general convex bodies $K$ in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding…
In this short note, we show that, in any given metric space, every Lipschitz open-map image of every subset of a given metric space whose boundary is Hausdorff-null is Hausdorff-measurable with respect to the same dimension. The main…
In this note we connect Sobolev estimates in the context of polynomial averages e.g. \[ \| \int_0^1 \prod_{k=1}^m f_k(x-t^k) \|_{1} \leq \text{Const} \cdot 2^{-\text{const} \cdot l} \prod_{i=1}^m \| f_k \|_m \] whenever some $f_i$ vanishes…
Given a compact basic semi-algebraic set we provide a numerical scheme to approximate as closely as desired, any finite number of moments of the Hausdorff measure on the boundary of this set. This also allows one to approximate interesting…
Let us consider a sphere $S^{n-1}$ of radius $r$ in $\mathbb{R}^n$, where we have fixed poles $N$ and $S$. Suppose that $K$ is a set in $\mathbb{R}^n$ containing a translated copy of each meridian (that is an $S^{n-2}$-sphere) of $S^{n-1}$.…
In this paper we derive aggregate separation bounds, named after Davenport-Mahler-Mignotte (\dmm), on the isolated roots of polynomial systems, specifically on the minimum distance between any two such roots. The bounds exploit the…
In recent works, a framework has been developed to describe (quantum) deformed, spherically symmetric and static black holes in four dimensions. The key idea of this so-called Effective Metric Description (EMD) is to parametrise…
We determine the Hausdorff limit-set of the Euclidean hypersurfaces with large $\lambda_1$ or small extrinsic radius. The result depends on the $L^p$ norm of the curvature that is assumed to be bounded a priori, with a critical behaviour…
It was recently shown that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a domain with an $n-1$ dimensional uniformly rectifiable boundary, in the presence of now well understood additional…
We prove that if the Hausdorff dimension of $E\subset\mathbb{R}^d$, $d\geq 2$ is greater than $\frac{d}{2}+\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $$\Delta_2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset\mathbb{R}^2$$ has…
The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set $A\subset \mathbb{R}^2$, if the Hausdorff dimension of $A$ is greater than $\frac{5}{4}$, then the distance set…
The Separating Hyperplane theorem is a fundamental result in Convex Geometry with myriad applications. Our first result, Random Separating Hyperplane Theorem (RSH), is a strengthening of this for polytopes. $\rsh$ asserts that if the…
A celebrated result due to Wolff says if $E$ is a compact subset of ${\Bbb R}^2$, then the Lebesgue measure of the distance set $\Delta(E)=\{|x-y|: x,y \in E \}$ is positive if the Hausdorff dimension of $E$ is greater than $\frac{4}{3}$.…
We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $\mathbf{P}(\mathbb{R}^{n}) \times…
We give a simple criterion on the set of probability tangent measures $\mathrm{Tan}(\mu,x)$ of a positive Radon measure $\mu$, which yields lower bounds on the Hausdorff dimension of $\mu$. As an application, we give an elementary and…
We consider the distance minimization problem to a real algebraic variety $X \subseteq \RR^n$ when the metric is induced by a polyhedral norm. Each point in the variety has a Voronoi cell whose geometry depends on the normal space at the…