Compact Convex Sets with Prescribed Facial Dimensions
While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the faces of general closed convex sets. We show that for any finite sequence of positive integers there exist compact convex sets which only have extreme points and faces with dimensions from this prescribed sequence. We also discuss another approach to dimensionality, considering the dimension of the union of all faces of the same dimension. We show that the questions arising from this approach are highly nontrivial and give examples of convex sets for which the sets of extreme points have fractal dimension.
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The ideas in this paper were motivated by the discussions that took place during a recent MATRIX program in approximation and optimisation held in July 2016. We are grateful to the MATRIX team for the enjoyable and productive research stay. We would also like to thank the two referees for their insightful corrections and remarks.
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