The emphasis will be on characterizing convex bodies K in n - space with either of the following two, mutually equivalent properties: (a) The set of all ordered n - simplexes inscribed in K achieving the maximum possible volume is homeomorphic to the orthogonal group O(n). (b) The set of all ordered n - simplexes circumscribed to K achieving the minimum possible volume is homeomorphic to the orthogonal group O(n). Trivial examples of such objects are the n - dimensional spherical balls, ellipsoids; there are several other shapes; they can all be characterized by an appropriate dynamical system that can be solved fairly explicitly for n = 2. There are no special preparation requirements beyond linear algebra and the classical groups of geomery (especially the groups of euclidean motions and of affine transformations)