Title:

Applying Tverberg type theorems to geometric problems

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 kneighbourly 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 {x ,..., Xk) C X, such that conv(A {x ,... Xk}) n conv(B {x ,... Xk}) = 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.
