Related papers: Width deviation of convex polygons

For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The minimum width…

We show that the distance in total variation between $(\mathrm{Tr}\ U, \frac{1}{\sqrt{2}}\mathrm{Tr}\ U^2, \cdots, \frac{1}{\sqrt{m}}\mathrm{Tr}\ U^m)$ and a real Gaussian vector, where $U$ is a Haar distributed orthogonal or symplectic…

For a polygon $P$ with holes in the plane, we denote by $\varrho(P)$ the ratio between the geodesic and the Euclidean diameters of $P$. It is shown that over all convex polygons with $h$~convex holes, the supremum of $\varrho(P)$ is between…

Erdos, Herzog and Piranian asked whether, for $n$ points in the plane with fixed diameter (maximum distance between points), an arrangement of a regular $n$-gon maximizes their product of all pairs of distances. Recently, it was discovered…

Let X be the random variable that counts the number of triangles in the random graph G(n,p). We show that for some absolute constant c, the probability that X deviates from its expectation by at least \lambda \var(X)^{1/2} is at most…

In 1990, Tiet\"av\"ainen showed that if the only information we know about a linear code is its dual distance $d$, then its covering radius $R$ is at most $\frac{n}{2}-(\frac{1}{2}-o(1))\sqrt{dn}$. While Tiet\"av\"ainen's bound was later…

We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number…

The minimum feature size of a crossing-free straight line drawing is the minimum distance between a vertex and a non-incident edge. This quantity measures the resolution needed to display a figure or the tool size needed to mill the figure.…

Let ${X_1,...,X_n}$ be i.i.d. random observations. Let $\mathbb{S}=\mathbb{L}+\mathbb{T}$ be a $U$-statistic of order $k\ge2$ where $\mathbb{L}$ is a linear statistic having asymptotic normal distribution, and $\mathbb{T}$ is a…

A \textit{Reinhardt polygon} is a convex $n$-gon that, for $n$ not a power of $2$, is optimal in three different geometric optimization problems, for example, it has maximal perimeter relative to its diameter. Some such polygons exhibit a…

To prove by probabilistic methods that every $(n-1)$-dimensional section of the unit cube in $R^n$ has volume at most $\sqrt 2$, K. Ball made essential use of the inequality $$ \frac{1}{\pi}\int_{-\infty}^{\infty} \left(\frac{\sin^2…

Consider the problem of fnding the smallest area convex $k$-gon containing $n\in\mathbb{N}$ congruent disks without an overlap. By using Wegner inequality in sphere packing theory we give a lower bound for the area of such polygons. For…

We study the approximability of general convex sets in $\mathbb{R}^n$ by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution $N(0,I_n)$ and the complexity of an…

A well-known result by Kant [Algorithmica, 1996] implies that n-vertex outerplane graphs admit embedding-preserving planar straight-line grid drawings where the internal faces are convex polygons in $O(n^2)$ area. In this paper, we present…

We investigate the location of the maximal gradient of the torsion function on certain non-symmetric planar domains. First, by establishing uniform estimates for convex narrow domains, we show that as a planar domain bounded by two graphs…

If an n-side unit regular polygon is divided into m equal sized parts, then what is the minimum length of the split line ${l_{m,n}}$? This problem has its practical application in real world. This paper proved that ${l_{2,3}} = \sqrt…

It has been a widely belief that for a planar convex domain with two coordinate axes of symmetry, the location of maximal norm of gradient of torsion function is either linked to contact points of largest inscribed circle or connected to…

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we show that the…

We study the problem of $(\epsilon,\delta)$-differentially private learning of linear predictors with convex losses. We provide results for two subclasses of loss functions. The first case is when the loss is smooth and non-negative but not…

Some 76 years ago P. Tur\'an was the first to establish lower estimations of the ratio of the maximum norm of the derivatives of polynomials and the maximum norm of the polynomials themselves on the interval I:=[-1,1] and on the unit disk…