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Towards a projective Upper/Lower bound theorem

Banff International Research Station for Mathematical Innovation and Discovery

The Upper and Lower bound Theorems in convex geometry deal, respectively, with the maximum and minimum number of facets that a convex d-dimensional polytope with n vertices can have. In this talk we will offer bounds for such numbers for what we refer to as the projective case of this problem. Namely, we are interested in the following: Problem 1: Given a set of $n$ points in general position $X \subset \R^{d}$ what is the maximum number of facets that $conv(T(X))$ can have, among all the possible permisible projective transformations $T$ of $X$? Problem 2: Given a set of $n$ points in general position $X \subset \R^{d}$ what is the maximum number of vertices that $conv(T(X))$ can have (as the support of the convex hull), among all the possible permisible projective transformations $T$? We define the \textbf{projective class of a set of points} $X \subset \R^{d}$ as the set of all possible point configurations that are the image of $X$ under a permissible projective transformation, and denote it $[X].$ Mimicking the polytope notation, let $f_k([X])$ be \textbf{the maximum number of $k$--faces} that the convex hull of a point configuration in the class $[X]$ can have, i.e. $f_k([X])= max_{Y \in [X]} \{f_k(conv(Y))\}.$ Finally we denote as $f_k(n),$ the minimum $f_k([X])$ over all the possible configurations of $n$ points, i.e. $$f_k(n)=min_{X \subset \R^d, |X|=n}\{f_k([X])\}.$$ With this notation, Problems \ref{P:affine} and \ref{P:affine_v} and can be interpreted as the task of finding the value of $f_{d-1}(n)$ and $f_{0}(n)$ respectively. Both problems are natural generalizations of the well-known McMullen's problem: What is the maximum $n$ such that any set of $n$ points in general position, $X \subset \R^{d},$ can de mapped by a permisible projective transformation onto the vertices of a convex polytope? These two problems are closely related. Moreover, Problem \ref{P:affine} has a direct application to the problem of counting the number of Radon partitions induced by a colouring.

Mathematics; Convex and discrete geometry; Group theory and generalizations; Discrete mathematics

video/mp4

Author affiliation: CONACYT-INFOTEC

2018-03-23

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64887

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/