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Many fundamental problems in computational geometry admit no algorithm running in $o(n \log n)$ time for $n$ planar input points, via classical reductions from sorting. Prominent examples include the computation of convex hulls, quadtrees,…

Let $G=(V(G), E(G))$ be an undirected graph with a measure function $\mu$ assigning non-negative values to subgraphs $H$ so that $\mu(H)$ does not exceed the clique cover number of $H$. When $\mu$ satisfies some additional natural…

A novel 2-D method for computing the convex hull of a sufficiently dense set of n integer points is introduced. The approach employs a ranking function that avoids sorting the points directly thus reducing the overall time complexity. The…

The convex hull of N independent random points chosen on the boundary of a simple polytope in R^n is investigated. Asymptotic formulas for the expected number of vertices and facets, and for the expectation of the volume difference are…

We study the following range searching problem: Preprocess a set $P$ of $n$ points in the plane with respect to a set $\mathcal{O}$ of $k$ orientations % , for a constant, in the plane so that given an $\mathcal{O}$-oriented convex polygon…

Given a bounded subanalytic submanifold of $\mathbb{R}^n$, possibly admitting singularities within its closure, we study the cohomology of $L^p$ differential forms having an $L^p$ exterior differential (in the sense of currents) and…

Finding a good classifier is a multiobjective optimization problem with different error rates and the costs to be minimized. The receiver operating characteristic is widely used in the machine learning community to analyze the performance…

$\def\DD{{{\bf \delta}}}\def\CH{{\mathop{\mathrm{ConvexHull}}}}\newcommand{\LL}{{\cal {L}}} \newcommand{\ZZ}{\mathbb{Z}} $ Given a convex body $C$ in the plane, its discrete hull is $C^0 = \CH( C \cap \LL )$, where $\LL = \ZZ \times \ZZ$ is…

The Separating Hyperplane theorem is a fundamental result in Convex Geometry with myriad applications. Our first result, Random Separating Hyperplane Theorem (RSH), is a strengthening of this for polytopes. $\rsh$ asserts that if the…

This paper studies Stochastic Shortest Path (SSP) problems in known and unknown environments from the perspective of convex optimisation. It first recalls results in the known parameter case, and develops understanding through different…

Let C be a compact and convex set in the plane that contains the origin in its interior, and let S be a finite set of points in the plane. The Delaunay graph DG_C(S) of S is defined to be the dual of the Voronoi diagram of S with respect to…

Fix any $\lambda\in\mathbb{C}$. We say that a set $S\subseteq\mathbb{C}$ is $\lambda$-$convex$ if, whenever $a$ and $b$ are in $S$, the point $(1-\lambda)a+\lambda b$ is also in $S$. If $S$ is also (topologically) closed, then we say that…

Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line…

Chan [JACM, 2010] gave a data structure for maintaining the convex hull of a dynamic set of 3D points under insertions and deletions, supporting extreme-point queries. Subsequent refinements by Kaplan, Mulzer, Roditty, Seiferth, and Sharir…

According to the classical Gauss-Lucas theorem all zeros of the derivative of a complex non-constant polynomial p lie in the convex hull of the zeros of p. It is proved that for a polynomial p of degree four with four different zeros…

For a given point set $S$ in a plane, we develop a distributed algorithm to compute the $\alpha-$shape of $S$. $\alpha-$shapes are well known geometric objects which generalize the idea of a convex hull, and provide a good definition for…

This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction…

Several issues in machine learning and inverse problems require to generate discrete data, as if sampled from a model probability distribution. A common way to do so relies on the construction of a uniform probability distribution over a…

This work develops a class of relaxations in between the big-M and convex hull formulations of disjunctions, drawing advantages from both. The proposed "P-split" formulations split convex additively separable constraints into P partitions…

We consider the problem of computing the time-convex hull of a point set under the general $L_p$ metric in the presence of a straight-line highway in the plane. The traveling speed along the highway is assumed to be faster than that off the…