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Register a new userVector spaces spanned by the angle sums of polytopes
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Kristin A. Camenga
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In this paper, we will describe the space spanned by the angle-sums of polytopes, recorded in the alpha-vector. We will consider the angles sums of simplices and the angles sums and face numbers of simplicial polytopes and general polytopes. We will construct families of polytopes whose angle sums span the spaces of polytopes defined by the Gram and Perles equations, analogues of the Euler and Dehn-Sommerville equations. We show that the dimensions of the affine span of the space of angle sums of simplices is floor[(d-1)/2] + 1, and that of the angle sums and face numbers of simplicial polytopes and general polytopes are d-1 and 2d-3 respectively.
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We will study the angle sums of polytopes, listed in the $alpha$-vector, working to exploit the analogy between the f-vector of faces in each dimension and the alpha-vector of angle sums. The Gram and Perles relations on the $alpha$-vector are analogous to the Euler and Dehn-Sommerville relations on the f-vector. First we describe the spaces spanned by the the alpha-vector and the $alpha$-f-vectors of certain classes of polytopes. Families of polytopes are constructed whose angle sums span the spaces of polytopes defined by the Gram and Perles equations. This shows that the dimension of the affine span of the space of angle sums of simplices is floor[(d-1)/2], and that of the combined angle sums and face numbers of simplicial polytopes and general polytopes are d-1 and 2d-3, respectively. Next we consider angle sums of polytopal complexes. We define the angle characteristic on the alpha-vector in analogy to the Euler characteristic. We show that the changes in the two correspond and that, in the case of certain odd-dimensional polytopal complexes, the angle characteristic is half the Euler characteristic. Finally, we consider spherical and hyperbolic polytopes and polytopal complexes. Spherical and hyperbolic analogs of the Gram relation and a spherical analog of the Perles relation are known, and we show the hyperbolic analog of the Perles relations in a number of cases. Proving this relation for simplices of dimension greater than 3 would finish the proof of this result. Also, we show how constructions on spherical and hyperbolic polytopes lead to corresponding changes in the angle characteristic and Euler characteristic.
Robertson (1988) suggested a model for the realization space of a convex d-dimensional polytope and an approach via the implicit function theorem, which -- in the case of a full rank Jacobian -- proves that the realization space is a manifold of dimension NG(P):=d(f_0+f_{d-1})-f_{0,d-1}, which is the natural guess for the dimension given by the number of variables minus the number of quadratic equations that are used in the definition of the realization space. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3),and Robertson claimed this to be true for all polytopes, Mnevs (1986/1988) Universality Theorem implies that it is not true in general: Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P) in general. In this paper we develop Jacobian criteria for the analysis of realization spaces. From these we get easily that for various large and natural classes of polytopes the realization spaces are indeed manifolds, whose dimensions are given by NG(P). However, we also identify the smallest polytopes where the dimension count (2) and thus Robertsons claim fails, among them the bipyramid over a triangular prism. For the property (1), we analyze the classical 24-cell: We show that the realization space has at least the dimension 48, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere.
It is an amazing and a bit counter-intuitive discovery by Micha Perles from the sixties that there are ``non-rational polytopes: combinatorial types of convex polytopes that cannot be realized with rational vertex coordinates. We describe a simple construction of non-rational polytopes that does not need duality (Perles ``Gale diagrams): It starts from a non-rational point configuration in the plane, and proceeds with so-called Lawrence extensions. We also show that there are non-rational polyhedral surfaces in 3-space, a discovery by Ulrich Brehm from 1997. His construction also starts from any non-rational point configuration in the plane, and then performs what one should call Brehm extensions, in order to obtain non-rational partial surfaces. These examples and objects are first mile stones on the way to the remarkable universality theorems for polytopes and for polyhedral surfaces by Mnev (1986), Richter-Gebert (1994), and Brehm (1997).
For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the $4$-dimensional cyclic polytopes with at least $8$ vertices---all of whose faces are inscribable---are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic $4$-polytopes with up to $7$ vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that $d$-dimensional cyclic polytopes with at least $d+4$ vertices are not circumscribable, and that no dual of a neighborly $4$-polytope with $8$ vertices, that is, no polytope with $f$-vector $(20,40,28,8)$, is inscribable.
Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitati
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