The cyclic polytope attains the maximum number of vertices in a hyperplane slice, and the paper determines the missing vertex counts, or gaps, for slices of hypercubes up to dimension seven.
Extremal sections and projections of certain convex bodies: a survey
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We survey results concerning sharp estimates on volumes of sections and projections of certain convex bodies, mainly $\ell_p$ balls, by and onto lower dimensional subspaces. This subject emerged from geometry of numbers several decades ago and since then has seen development of a variety of probabilistic and analytic methods, showcased in this survey.
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On the Number of Vertices in a Hyperplane Section of a Polytope
The cyclic polytope attains the maximum number of vertices in a hyperplane slice, and the paper determines the missing vertex counts, or gaps, for slices of hypercubes up to dimension seven.