Webb, Simon Peter; (1996) Central slices of the regular simplex. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

In this thesis we investigate different methods of proving best upper bounds for the volumes of central sections of the regular n-dimensional simplex. In Chapter 1 we show, using probabilistic methods, that the 1-codimensional central sections with maximal volume are exactly those sections that contain n - 1 of the vertices of the simplex. The proof uses results about logarithmically concave functions on R. We note that there are both similarities and differences between this proof and that for the case of the n-dimensional cube, and we also give an intriguing reinterpretation of the result involving interpolation. In Chapter 2 we examine the possibility of extending the 1-codimensional result of Chapter 1 to sections of any dimension. We show that the problem will reduce to a question about the position of the centroid of central slices of regular simplices in one dimension lower. In Chapter 3 we show that the maximal 2-dimensional central slices of the regular simplex are those that contain 2 of the vertices. We prove this by obtaining best upper bounds on the volumes of maximal ellipsoids in central slices of the simplex. The proof involves making estimates on the determinants of matrices of the form [equation] is a sequence of vectors in Rk. Chapter 4 is a discussion of how our new results compare with those of P.Filliman, who gave conditions that must be satisfied by critical central sections (with respect to volume) of the regular simplex.

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