The number of distinct distances from a vertex of a convex polygon

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Two subsets A,B of an n-element ground set X are said to be crossing, if none of the four sets AB, A\B, B\A and X(AB) are empty. It was conjectured by Karzanov and Lomonosov forty years ago that if a family F of subsets of X does not contain k pairwise cr ...

Cao and Yuan obtained a Blichfeldt-type result for the vertex set of the edge-to-edge tiling of the plane by regular hexagons. Observing that the vertex set of every Archimedean tiling is the union of translates of a fixed lattice, we take a more general v ...

The problem of allocating the closed-loop poles of linear systems in specific regions of the complex plane defined by discrete time-domain requirements is addressed. The resulting non-convex set is inner-approximated by a convex region described with linea ...

A straight-line drawing of a graph G is a mapping which assigns to each vertex a point in the plane and to each edge a straight-line segment connecting the corresponding two points. The rectilinear crossing number of a graph G, (cr) over bar (G), is the mi ...

We present new techniques to analyze natural local search algorithms for several variants of the max-sum diversification problem which, in its most basic form, is as follows: given an n-point set X subset of R-d and an integer k, select k points in X so th ...

Given n continuous open curves in the plane, we say that a pair is touching if they have finitely many interior points in common and at these points the first curve does not get from one side of the second curve to its other side. Otherwise, if the two cur ...

A graph G is a diameter graph in R-d if its vertex set is a finite subset in R-d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in R-4 contains the complete subgraph K on five vertices, then any ...

We show that for m points and n lines in R-2, the number of distinct distances between the points and the lines is Omega(m(1/5)n(3/5)), as long as m(1/2)

We develop a primal-dual convex minimization framework to solve a class of stochastic convex three-composite problem with a linear operator. We consider the cases where the problem is both convex and strongly convex and analyze the convergence of the propo ...

Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, for example, as lengths of catheti of right-angled triangles (defined via so-called quasiorthogonality). These triangles have two ...

Let S be a set of n points in R-2 contained in an algebraic curve C of degree d. We prove that the number of distinct distances determined by S is at least c(d)n(4/3), unless C contains a line or a circle. We also prove the lower bound c(d)' min{m(2/3)n(2/ ...

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ordinary circles. Moreover, we determine the exact minimu ...

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\ge 3

, we define

e(n)

to be the minimum positive integer such that an ...

The present thesis deals with problems arising from discrete mathematics, whose proofs make use of tools from algebraic geometry and topology. The thesis is based on four papers that I have co-authored, three of which have been published in journals, and o ...

The number of distinct distances from a vertex of a convex polygon

Chat with Graph Search

On the Size of K-Cross-Free Families

A note on discrete lattice-periodic sets with an application to Archimedean tilings

Discrete-time regional pole-placement using convex approximations: Theory and application to a boost converter

Approximating the rectilinear crossing number

An Improved Analysis of Local Search for Max-Sum Diversification

Many touchings force many crossings

On Simplices in Diameter Graphs in R-4

Distinct distances between points and lines

Stochastic Three-Composite Convex Minimization with a Linear Operator

New Moduli For Banach Spaces

Distinct Distances on Algebraic Curves in the Plane

On Sets Defining Few Ordinary Circles

On some algebraic and extremal problems in discrete geometry

Algebraic and topological methods in combinatorics