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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). Green open access

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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)
Title: Applying Tverberg type theorems to geometric problems
Identifier: PQ ETD:593652
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: 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 classification: UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1446307
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