We consider sets L = {l(1),..., l(n)} of n labeled lines in general position in R-3, and study the order types of point sets {p(1),..., p(n)} that stem from the intersections of the lines in L with (directed) planes Pi, not parallel to any line of L, that ...
In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer n≥3, we define e(n) to be the minimum positive integer such that an ...
We show that the lines of every arrangement of n lines in the plane can be colored with O(root n/log n) colors such that no face of the arrangement is monochromatic. This improves a bound of Bose et al. by a circle minus(root/log n) factor. Any further imp ...
Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is 13n/36 - O(1). I ...
Iterative models are widely used today in CAD. They allow, with a limited number of parameters, to represent relatively complex forms through a subdivision algorithm. There is a wide variety of such models (Catmull-Clark, Doo-Sabin, L-Systems...). Most ite ...
We present a detailed study of the low-energy excitations of two existing finite-size realizations of the planar kagome Heisenberg antiferromagnet on the sphere: the cuboctahedron and the icosidodecahedron. After highlighting a number of special spectral f ...
In this paper, we prove a strengthening of the generic vanishing result in characteristic p > 0 given in Hacon and Patakfalvi (Am J Math 138(4):963-998, 2016). As a consequence of this result, we show that irreducible Theta divisors are strongly F-regular ...
Let ES(n) denote the minimum natural number such that every set of ES(n) points in general position in the plane contains n points in convex position. In 1935, ErdAs and Szekeres proved that ES. In 1961, they obtained the lower bound , which they conjectur ...
Minkowski sums are a very simple geometrical operation, with applications in many different fields. In particular, Minkowski sums of polytopes have shown to be of interest to both industry and the academic world. This thesis presents a study of these sums, ...
A polyhedral subdivision of a d-dimensional point configuration A is k-regular if it is projected from the boundary complex of a polytope with dimension at most d+k. Call γk(A) the subgraph induced by k-regular triangulations in the flip-graph of A. Gel’fa ...
