Regular Convex Bodies

Abstract

It is well-known that a convex polytope is regular if and only if the action of its symmetry group on the set of complete flags is transitive. We use this criterion to extend the notion of regularity from convex polytopes to convex bodies in general.To achieve this extension, we introduce the notion of maximality for flags, which reduces to completeness for convex polytopes. A convex body whose symmetry group acts transitively on the set of maximal flags (and which satisfies a further condition which is automatic for convex polytopes) will be called a regular solid.It is easy to show that if B is a regular solid, then so is its polar B*. Moreover, all regular solids are perfect in the sense of [7]. Our main structure theorem concerns a projection n: ¥ R -> 0> R from the set Sf R of all regular solids onto the set 2P R of all regular polytopes. Conversely, a construction of Kostant [5] associates either one or two regular nonpolytopes to certain regular polytopes. The relation between n and Kostant's construction is not yet fully elucidated, but does at least suggest conjectures on the classification of regular solids, and of perfect solids in general.We also explore the less stringent notions of /-transitivity and complete transitivity, finding conditions under which these properties are preserved under product and coproduct operations.The work is organised in three sections. Section 1 examines convex bodies in general, introducing some ideas and constructions that are needed later. Section 2 deals with the notion of regularity, and Section 3 with perfection and transitivity.

Strongly symmetric spectral convex bodies are Jordan algebra state spaces

“…It is a compact group. (This is often called Ω's symmetry group but (mainly for consistency with [7,8]) we'll reserve that term for use in a slightly different context to be introduced later.) We'll often denote it with the letter K. In the GPT literature, its elements are often called reversible transformations.…”

“…{0}). 6 A cone V + is said to be perfect if V can be equipped with an inner product such that every face F of V + , including V + itself, is equal to its dual with respect to the restriction of the inner product to lin F. 7 In the literature on general probabilistic theories, systems are sometimes modeled using an explicit specification of a proper subset of the sets of effects and of measurements, or positive maps, subject to reasonable conditions, called the "allowed" or "physical" effects, measurements, or positive maps. When the full set of effects is allowed, the system is said to satisfy the "no-restriction hypothesis".…”

“…[5]), so the new result here is the converse. It is obtained by showing that compact convex sets in finite dimension satisfying spectrality and strong symmetry are (up to affine isomorphism) a subset of the regular convex bodies, defined in [7] as bodies whose symmetry group acts transitively on maximal chains of faces, and using the classification of those convex bodies in [8] to verify that this subset consists precisely of the abovementioned classes of sets.…”

Dynamics and geometry in forced symmetry breaking: a tetrahedral example

“…There is a natural way to represent an orbit in V 2 of maximal isotropy type as the image of a 2 : 1 immersion of a 2-sphere into V 2 which describes the orbit in classical terms as a Veronese surface [2]. This construction has been used to valuable effect in the study of regular convex bodies [15,34], and in the analysis of symmetry detectives [6]. It turns out to be an extremely useful tool for understanding vector fields on orbits in V 2 and their symmetry properties.…”

Orbitopes

“…Orbitopes of finite groups are highly symmetric convex polytopes which include the platonic solids, permutahedra, Birkhoff polytopes, and other favorites from Ziegler's text book [38], as well as the Coxeter orbihedra studied by McCarthy, Ogilvie, Zobin, and Zobin [25]. Farran and Robertson's regular convex bodies [11] are orbitopal generalzations of regular polytopes, which were classified by Madden and Robertson [24]. Orbitopes for compact Lie groups, such as SO(n), have appeared in investigations ranging from protein structure prediction [23] and quantum information [2] to calibrated geometries [16].…”