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Register a new userThe Regge symmetry, confocal conics, and the Schlafli formula
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The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry.
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We prove that if a framework of a graph is neighborhood affine rigid in $d$-dimensions (or has the stronger property of having an equilibrium stress matrix of rank $n-d-1$) then it has an affine flex (an affine, but non Euclidean, transform of space that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.
The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this result to convergence by monomorphisms, so that symmetries can only increase on passing to the limit.
At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes many known inequalities as special cases. The aim of this note is to give new proofs of the Alexandrov-Fenchel inequality and of its matrix counterpart, Alexandrovs inequality for mixed discriminants, that appear conceptually and technically simpler than earlier proofs and clarify the underlying structure. Our main observation is that these inequalities can be reduced by the spectral theorem to certain trivial `Bochner formulas.
In this work we study the issue of geodesic extendibility on complete and locally compact metric length spaces. We focus on the geometric structure of the space $(Sigma (X),d_H)$ of compact balls endowed with the Hausdorff distance and give an explicit isometry between $(Sigma (X),d_H)$ and the closed half-space $ Xtimes mathbb{R}_{ge 0}$ endowed with a taxicab metric. Among the applications we establish a group isometry between $mbox{Iso} (X,d)$ and $mbox{Iso} (Sigma (X),d_H)$ when $(X,d)$ is a Hadamard space.
In a seminal paper Volumen und Oberflache (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the extremals of these inequalities was already emphasized by Minkowski himself, but has to date only been resolved in special cases. In this paper, we completely settle the extremals of Minkowskis quadratic inequality, confirming a conjecture of R. Schneider. Our proof is based on the representation of mixed volumes of arbitrary convex bodies as Dirichlet forms associated to certain highly degenerate elliptic operators. A key ingredient of the proof is a quantitative rigidity property associated to these operators.
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