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In this paper we present several results on the expected complexity of a convex hull of $n$ points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of $n$ points,…

In this article we prove a conjecture of Bezdek, Brass, and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of n unit hypercubes in the d-dimensional Euclidean space. For d=2 we enumerate the…

A new line of research on the lasso exploits the beautiful geometric fact that the lasso fit is the residual from projecting the response vector $y$ onto a certain convex polytope. This geometric picture also allows an exact geometric…

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…

We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree…

In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved (\cite{Sa}, \cite{CaSoA}). However, the question for multidimensional Lorentz spaces is still open. In this paper,…

We consider bounded integer knapsacks where the weights and variable upper bounds together form a superincreasing sequence. The elements of this superincreasing knapsack are exactly those vectors that are lexicographically smaller than the…

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…

We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the…

For a large class of tilings, including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull of such a tiling inherits a minimal lamination structure with flat leaves and a transversal…

We prove that a general class of measures, which includes $\log$-concave measures, is $\frac{1}{n}$-concave according to the terminology of Borell, with additional assumptions on the measures or on the sets, such as symmetries. This…

Consider a measurable space with an atomless finite vector measure. This measure defines a mapping of the $\sigma$-field into an Euclidean space. According to the Lyapunov convexity theorem, the range of this mapping is a convex compactum.…

A set S of vertices of a connected graph G is convex, if for any pair of vertices u; v 2 S, every shortest path joining u and v is contained in S . The convex hull CH(S) of a set of vertices S is defined as the smallest convex set in G…

We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary…

One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some…

We define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the Lorentzian setting, this allows us to define a geometric dimension - akin to the Hausdorff dimension for metric spaces - that…

We prove that for certain families of semi-algebraic convex bodies in 3 dimensions, the convex hull of $n$ disjoint bodies has $O(n\lambda_s(n))$ features, where $s$ is a constant depending on the family: $\lambda_s(n)$ is the maximum…

We consider the problem of reporting convex hull points in an orthogonal range query in two dimensions. Formally, let $P$ be a set of $n$ points in $\mathbb{R}^{2}$. A point lies on the convex hull of a point set $S$ if it lies on the…

In this paper a measure of non-convexity for a simple polygonal region in the plane is introduced. It is proved that for "not far from convex" regions this measure does not decrease under the Minkowski sum operation, and guarantees that the…

This article describes a method to compute successive convex approximations of the convex hull of a set of points in R^n that are the solutions to a system of polynomial equations over the reals. The method relies on sums of squares of…