CHAPTER 1.11 - Special Convex Bodies

Abstract

This chapter discusses special convex bodies like simplices, ellipsoids, centrally symmetric bodies, and bodies of constant width. A simplex in Ed is defined as the convex hull of d + 1 affinely independent points. Leichtweiss used the infimum of the quotients of circumradius and inradius over classes of affinely equivalent convex bodies, Santaló and Eggleston used the area and a certain width integral of a plane convex body in order to characterize simplices. For studying ellipsoids from the view point of convexity one has a good starting point in the surveys of Bonnesen and Fenchel, Gruber and Höbinger, and Petty, which summarize results on characterizations of ellipsoid by means of sections and projections; by extremal properties; using the group of projectivities or subgroups thereof; and also on ellipsoids in Minkowski geometry and Hilbert geometry. Furthermore, there are a lot of geometric inequalities yielding ellipsoids as extremal bodies. Also, approximation of convex bodies by convex polytopes may lead to ellipsoids as extremal bodies. Ellipsoids may also be characterized by the product of mean width and surface area or by the quotient of inradius and circumradius. Characterizations of Euclidean spaces or, in infinite dimensions, of inner product spaces also give characterizations of ellipsoids. Also, Klee and Bolker have characterized centrally symmetric polytopes by intersection and projection properties.