Related papers: A quantitative version of the Steinhaus theorem

A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite binary sequence {\omega} and an effectively open set A of measure less than 1, some tail of {\omega} is not in A. We first prove several results in the same spirit and…

It is shown that, given a point $x\in\mathbbm{R}^d$, $d\ge 2$, and open sets $U_1,...,U_k$ containing $x$, any convex combination of the harmonic measures for $x$ with respect to $U_n$, $1\le n\le k$, is the limit of a sequence of harmonic…

Assume that $k \le d$ is a positive integer and $\C$ is a finite collection of convex bodies in $\R^d$. We prove a Helly type theorem: If for every subfamily $\C^*\subset \C$ of size at most $\max \{d+1,2(d-k+1)\}$ the set $\bigcap \C^*$…

It is well known that axiom of choice implies the existence of non-measurable sets for Lebesgue's measure on R as well as the existence of "paradoxical" decompositions of the unit ball of R^3 (Banach-Tarski). This is generally interpreted…

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the dbar-method, avoids moving arguments and gives much stronger results. In particular, it is proved that if X and Y are connected smooth…

We define an $S_\bullet$-construction for squares categories, and introduce a class of squares categories we call "proto-Waldhausen" which capture the properties required for the $S_\bullet$-construction to model the K-theory space. The…

A theorem of Tietze and Nakamija, from 1928, asserts that if a subset X of R^n is closed, connected, and locally convex, then it is convex. We give an analogous "local to global convexity" theorem when the inclusion map of X to R^n is…

Let $E \subset \mathbb R^d$, $d \ge 2$, be compact, and let $\phi(x,y)$ be a smooth function satisfying the Phong--Stein rotational curvature condition on $\{\phi(x,y)=1\}$. We prove that if $\dim_{\mathcal H}(E)>1$, then $$…

We study properties of convex hulls of (co)adjoint orbits of compact groups, with applications to invariant theory and tensor product decompositions. The notion of partial convex hulls is introduced and applied to define two numerical…

For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function…

The Wu--Yau theorem asserts that a compact K\"ahler manifold with negative holomorphic sectional curvature admits a cohomologous metric with negative Ricci curvature. We introduce a conjectural positive analog of the Wu--Yau theorem and…

Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $\mu$ with the property that $$ \int_{X} d(x, y) d\mu(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist,…

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote…

Gr\"unbaum's inequality guarantees that the centroid of a convex body has halfspace depth at least $1/e$: every halfspace containing the centroid captures at least a $1/e$ fraction of the body's volume. For mixed-integer convex sets…

Gaussian processes can be considered as subsets of a standard Hilbert space, but the geometric understanding that would relate the size of a set with the size of its convex hull is still lacking. In this work, we adopt a geometric approach…

The Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This…

We observe that the maximal open set of constant curvature k in a Riemannian manifold with curvature bounded below or above by k has a convexity type property, which we call "two-convexity". This statement is used to prove a number of…

This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact…

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other…

Let $C$ be a closed convex cone in ${\mathbb R}^n$, pointed and with interior points. We consider sets of the form $A=C\setminus A^\bullet$, where $A^\bullet\subset C$ is a closed convex set. If $A$ has finite volume (Lebesgue measure),…