Related papers: Volume of tubes, non polynomial behavior

The "Mahler volume" is, intuitively speaking, a measure of how "round" a centrally symmetric convex body is. In one direction this intuition is given weight by a result of Santalo, who in the 1940s showed that the Mahler volume is…

Let $X_1,\ldots,X_N$, $N>n$, be independent random points in $\mathbb{R}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more…

In this article, we study the (d-1)-volume and the covering numbers of the medial axis of a compact set of the Euclidean d-space. In general, this volume is infinite; however, the (d-1)-volume and covering numbers of a filtered medial axis…

Let $f = (f_1,\ldots,f_m) : \R^n \longrightarrow \R^m$ be a polynomial map; $G^f(r) = \{x\in\R^n : |f_i(x)| \leq r,\ i =1,\ldots, m\}$. We show that if $f$ satisfies the Mikhailov - Gindikin condition then \begin{itemize} \item[(i)]…

In this short note, we consider the question of determining the asymptotics of the volume function near the boundary of the pseudoeffective cone on compact K\"ahler manifolds. We solve the question in a number of cases -- in particular, we…

Federer's characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension one. In complete…

We establish new linear and trilinear bounds for collections of tubes in $\mathbb{R}^4$ that satisfy the polynomial Wolff axioms. In brief, a collection of $\delta$-tubes satisfies the Wolff axioms if not too many tubes can be contained in…

We study open point sets in Euclidean spaces $\mathbb{R}^d$ without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density zero. We are interested in how…

When placed in a crowded environment, a semi-flexible tube is forced to fold so as to make a more compact shape. One compact shape that often arises in nature is the tight helix, especially when the tube thickness is of comparable size to…

The Mahler measure of the polynomials $t(x^m-1) y - (x^n-1) \in \dC[x,y]$ is essentially the sum of volumes of a certain collection of ideal hyperbolic polyhedra in $\HH^3$, which can be determined a priori as a function on the parameter…

Let $P$ be a bounded polyhedron defined as the intersection of the non-negative orthant ${\Bbb R}^n_+$ and an affine subspace of codimension $m$ in ${\Bbb R}^n$. We show that a simple and computationally efficient formula approximates the…

Brightwell, van den Heuvel and Stougie proved that the diameter of an $m \times n$ transportation polytope is at most $8(m+n-2)$, a factor of eight away from the Hirsch Conjecture. This bound was improved to $3(m+n-1)$ by Hurkens. We…

We study the $\delta$-discretized Szemer\'edi-Trotter theorem and Furstenberg set problem. We prove sharp estimates for both two problems assuming tubes satisfy some spacing condition. For both two problems, we construct sharp examples that…

The aim of this note is firstly to give a new brief proof of classical Bochner's Tube Theorem (1938) by making use of K. Oka's Boundary Distance Theorem (1942), showing directly that two points of the envelope of holomorphy of a tube can be…

We develop a new method to estimate the area, and more generally the intrinsic volumes, of a compact subset $X$ of $\mathbb{R}^d$ from a set $Y$ that is close in the Hausdorff distance. This estimator enjoys a linear rate of convergence as…

For $n \geq 2$ we construct a measurable subset of the unit ball in $\mathbb{R}^n$ that does not contain pairs of points at distance 1 and whose volume is greater than $(1/2)^n$ times the volume of the ball. This disproves a conjecture of…

We study sets of $\delta$ tubes in $\mathbb{R}^3$, with the property that not too many tubes can be contained inside a common convex set $V$. We show that the union of tubes from such a set must have almost maximal volume. As a consequence,…

We give upper bounds for volume of sublevel sets of real polynomials. Our method is to combine a version of global Lojasiewicz inequality with some well known estimate on volume of tubes around real algebraic sets. Some applications to…

A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$. The formula is known to hold when $T \le 3k-4$, for some small range over $3k-4$…

Let $M^n$ be a complete, open Riemannian manifold with $\Ric \geq 0$. In 1994, Grigori Perelman showed that there exists a constant $\delta_{n}>0$, depending only on the dimension of the manifold, such that if the volume growth satisfies…