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## Applying Tverberg type theorems to geometric problems

Colin, GN; (2007) Applying Tverberg type theorems to geometric problems. Doctoral thesis , UCL (University College London).   Preview Text Colin.Garcia.Natalia_thesis.Redacted.pdf Download (5MB) | Preview

## Abstract

In this thesis three main problems are studied. The first is a generalization of a well known question by P. McMullen on convex polytopes: 'Determine the largest number v(d, k) such that any set of u(d, k) points lying in general position in M.d can be mapped, by a permissible projective transformation, onto the vertices of a k-neighbourly polytope.7 Bounds for u(d, k) are obtained. The upper bound is attained using oriented matroid techniques. The lower bound is proved indirectly, by considering a partition problem equivalent to McMullen's question. The core partition problem, mentioned above, can be modified in the following manner: 'Let X be a set of n points in general position in Rd then, what is the minimum k such that for all A, B partition of X there is always a set {lcub}x ,..., Xk) C X, such that conv(A {lcub}x ,... Xk{rcub}) n conv(B {lcub}x ,... Xk{rcub}) = 0' For this question, through an asymptotical analysis, a relationship between the number of points in the set (n) , and the number to be removed (k) , is shown. Finally, another problem in convex polytopes proposed by von Stengel is considered: 'Consider a polytope, V, in dimension d with 2d facets, which is simple. Two vertices form a complementary pair, (x,y), if every facet of V is incident with x or y. The d cube has 2d l complementary vertex pairs. Is this the maximal number among the simple d polytopes with 2d facets', It is shown that the conjecture stated above holds up to dimension seven and extra conditions, under which the theorem holds in general, are exposed. A nice interpretation of von Stengel's question, in terms of coloured Radon partitions, is also introduced.

Type: Thesis (Doctoral) Applying Tverberg type theorems to geometric problems PQ ETD:593652 An open access version is available from UCL Discovery English Thesis digitised by ProQuest. Third party copyright material has been removed from the ethesis. Images identifying individuals have been redacted or partially redacted to protect their identity. UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics https://discovery.ucl.ac.uk/id/eprint/1446307 View Item