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Manor Mendel

Publications and source records attributed to Manor Mendel.

At least 19 recordsLinked to original sources
arXiv2026

Regular Ultrametric Skeletons

The ultrametric skeleton theorem extracts from every compact metric probability space a subset of ultrametric distortion $O(1/\varepsilon)$ that carries a measure whose balls are controlled by the $(1-\varepsilon)$-power of the original measure on dilated concentric balls. We prove a two-sided version for arbitrary compact metric spaces: for every ball centered on the skeleton, the skeleton measure also has a lower bound in terms of the original measure on a smaller nonconcentric ball contained in it. We also give a short proof of the original skeleton theorem and improve the dilation of its control balls from $\exp(O(1/\varepsilon^2))$ to $O(1/\varepsilon)$.

math.MG↗
arXiv2025

An optimal algorithm for average distance in typical regular graphs

We design a deterministic algorithm that, given $n$ points in a \emph{typical} constant degree regular~graph, queries $O(n)$ distances to output a constant factor approximation to the average distance among those points, thus answering a question posed in~\cite{MN14}. Our algorithm uses the method of~\cite{MN14} to construct a sequence of constant degree graphs that are expanders with respect to certain nonpositively curved metric spaces, together with a new rigidity theorem for metric transforms of nonpositively curved metric spaces. The fact that our algorithm works for typical (uniformly random) constant degree regular graphs rather than for all constant degree graphs is unavoidable, thanks to the following impossibility result that we obtain: For every fixed $k\in \N$, the approximation factor of any algorithm for average distance that works for all constant degree graphs and queries $o(n^{1+1/k})$ distances must necessarily be at least $2(k+1)$. This matches the upper bound attained by the algorithm that was designed for general finite metric spaces in~\cite{BGS}. Thus, any algorithm for average distance in constant degree graphs whose approximation guarantee is less than $4$ must query $\Omega(n^2)$ distances, any such algorithm whose approximation guarantee is less than $6$ must query $\Omega(n^{3/2})$ distances, any such algorithm whose approximation guarantee less than $8$ must query $\Omega(n^{4/3})$ distances, and so forth, and furthermore there exist algorithms achieving those parameters.

cs.DS↗
arXiv2022

A simple proof of Dvoretzky-type theorem for Hausdorff dimension in doubling spaces

The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any $0<β<α$, any compact metric space $X$ of Hausdorff dimension $α$ contains a subset which is biLipschitz equivalent to an ultrametric and has Hausdorff dimension at least $β$. In this note we present a simple proof of the ultrametric skeleton theorem in doubling spaces using Bartal's Ramsey decompositions [Bartal 2021]. The same general approach is also used to answer a question of Zindulka [Zindulka 2020] about the existence of "nearly ultrametric" subsets of compact spaces having full Hausdorff dimension.

math.MG↗
arXiv2022

Dvoretzky-type theorem for Ahlfors regular spaces

It is proved that for any $0<β<α$, any bounded Ahlfors $α$-regular space contains a $β$-regular compact subset that embeds biLipschitzly in an ultrametric with distortion at most $O(α/(α-β))$. The bound on the distortion is asymptotically tight when $β\to α$. The main tool used in the proof is a regular form of the ultrametric skeleton theorem.

math.MG↗
arXiv2020

Reliable Spanners for Metric Spaces

A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes can only cause a small damage in the remaining graph in terms of the dilation, that is, the spanner property is maintained for almost all nodes in the residual graph. Constructions of reliable spanners of near linear size are known in the low-dimensional Euclidean settings. Here, we present new constructions of reliable spanners for planar graphs, trees and (general) metric spaces.

cs.CG↗
arXiv2019

A Simple proof of Johnson-Lindenstrauss extension theorem

Johnson and Lindenstrauss proved that any Lipschitz mapping from an $n$-point subset of a metric space into Hilbert space can be extended to the whole space, while increasing the Lipschitz constant by a factor of $O(\sqrt{\log n})$. We present a simplification of their argument that avoids dimension reduction and the Kirszbraun theorem.

math.MG↗
arXiv2019

Nonpositive curvature is not coarsely universal

We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni, Naor and Neiman (2015). We establish this statement by proving that a metric space which is $q$-barycentric for some $q\in [1,\infty)$ has metric cotype $q$ with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the $\ell_\infty$ grid $[m]_\infty^n=(\{1,\ldots,m\}^n,\|\cdot\|_\infty)$ into $\ell_q$ for all $q\in (2,\infty)$, from which we deduce the following discrete converse to the fact that $\ell_\infty^n$ embeds with distortion $O(1)$ into $\ell_q$ for $q=O(\log n)$. A rigidity theorem of Ribe (1976) implies that for every $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ such that if $[m]_\infty^n$ embeds into $\ell_q$ with distortion $O(1)$, then $q$ is necessarily at least a universal constant multiple of $\log n$. Ribe's theorem does not give an explicit upper bound on this $m$, but by the work of Bourgain (1987) it suffices to take $m=n$, and this was the previously best-known estimate for $m$. We show that the above discretization statement actually holds when $m$ is a universal constant.

math.MG↗
arXiv2017

A relation between finitary Lipschitz extension moduli

This short note contains an elementary observation in response to the recent posting arXiv:1707.06593v1, which studies the Lipschitz extension modulus to $n$ additional points. We bound this modulus in terms of the well-studied Lipschitz extension modulus from $n$ points to the entire ambient metric space, thus making it possible to quote the available literature to improve some of the bounds obtained in arXiv:1707.06593v1.

math.MG↗
arXiv2014

Expanders with respect to Hadamard spaces and random graphs

It is shown that there exists a sequence of 3-regular graphs $\{G_n\}_{n=1}^\infty$ and a Hadamard space $X$ such that $\{G_n\}_{n=1}^\infty$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with respect to $X$. This answers a question of \cite{NS11}. $\{G_n\}_{n=1}^\infty$ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.

math.MG↗
arXiv2013

Spectral calculus and Lipschitz extension for barycentric metric spaces

The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT(0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.

math.MG↗
arXiv2013

Nonlinear spectral calculus and super-expanders

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.

math.MG↗
arXiv2012

A node-capacitated Okamura-Seymour theorem

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal c > 0, if the node cut conditions are satisfied, then one can simultaneously route a c-fraction of all the demands. This answers an open question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of multi-commodity polymatroid networks introduced by Chekuri, et. al. Our approach employs a new type of random metric embedding in order to round the convex programs corresponding to these more general flow problems.

math.CO↗
arXiv2012

Ultrametric subsets with large Hausdorff dimension

It is shown that for every $\e\in (0,1)$, every compact metric space $(X,d)$ has a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/\e)$, and $$\dim_H(S)\ge (1-\e)\dim_H(X),$$ where $\dim_H(\cdot)$ denotes Hausdorff dimension. The above $O(1/\e)$ distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.

math.MG↗
arXiv2011

Ultrametric skeletons

We prove that for every $ε\in (0,1)$ there exists $C_ε\in (0,\infty)$ with the following property. If $(X,d)$ is a compact metric space and $μ$ is a Borel probability measure on $X$ then there exists a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/ε)$, and a probability measure $ν$ supported on $S$ satisfying $ν(B_d(x,r))\le (μ(B_d(x,C_εr))^{1-ε}$ for all $x\in X$ and $r\in (0,\infty)$. The dependence of the distortion on $ε$ is sharp. We discuss an extension of this statement to multiple measures, as well as how it implies Talagrand's majorizing measures theorem.

math.MG↗
arXiv2011

A note on dichotomies for metric transforms

We show that for every nondecreasing concave function w:R+ --> R+ with w(0)=0, either every finite metric space embeds with distortion arbitrarily close to 1 into a metric space of the form (X,w o d) for some metric d on X, or there exists a=a(w)>0 and n_0=n_0(w)\in N such that for all n>n_0, any embedding of {0,...,n} into a metric space of the form (X,w o d) incurs distortion at least n^a.

math.MG↗
arXiv2011

Metric Cotype

We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L_p coarsely or uniformly embeds into L_q. We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question posed by Arora, Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matousek.

math.FA↗