Mathematics

Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles. In this paper, we extend the Connelly-Gortler method to allow circles to overlap by angles up to π/2 . As a result, we obtain a new proof of the general Koebe-Andre'ev-Thurston theorem for disk packings on S 2 with overlaps and a numerical algorithm for computing them. Our development makes use of the correspondence between circles and disks on S 2 and hyperplanes and half-spaces in the 4-dimensional Minkowski spacetime R 1,3 , which we illuminate in a preliminary section. Using this view we generalize a notion of convexity of circle polyhedra that has recently been used to prove the global rigidity of certain circle packings. Finally, we use this view to show that all convex circle polyhedra are infinitesimally rigid, generalizing a recent related result.

In non-Euclidean geometry, there are several known correspondings to Chapple-Euler Theorem. This remark shows that those results yield expressions corredponding to the well-known formula d= R(R??r) ??????????????????.

In this note we compare two ways of measuring the n -dimensional "flatness" of a set S??R d , where n?�N and d>n . The first one is to consider the classical Reifenberg-flat numbers α(x,r) ( x?�S , r>0 ), which measure the minimal scaling-invariant Hausdorff distances in B r (x) between S and n -dimensional affine subspaces of R d . The second is an `intrinsic' approach in which we view the same set S as a metric space (endowed with the induced Euclidean distance). Then we consider numbers a(x,r) 's, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at x of radius r in S and the n -dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers a(x,r) 's behaves as the square of the numbers α(x,r) 's. Moreover we show how this result finds application in extending the Cheeger-Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones' numbers β 's (i.e. the one-sided version of the numbers α 's).

The famous Minkowski inequality provides a sharp lower bound for the mixed volume V(K,M[n−1]) of two convex bodies K,M⊂ R n in terms of powers of the volumes of the individual bodies K and M . The special case where K is the unit ball yields the isoperimetric inequality. In the plane, Betke and Weil (1991) found a sharp upper bound for the mixed area of K and M in terms of the perimeters of K and M . We extend this result to general dimensions by proving a sharp upper bound for the mixed volume V(K,M[n−1]) in terms of the mean width of K and the surface area of M . The equality case is completely characterized. In addition, we establish a stability improvement of this and related geometric inequalities of isoperimetric type.

We prove a coarea-type inequality for a continuously Pansu differentiable function acting between two Carnot groups endowed with homogeneous distances. We assume that the level sets of the function are uniformly lower Ahlfors regular and that the Pansu differential is everywhere surjective.

The aim of this note is to give a sufficient condition for pairs of functions to have a convex separator when the underlying structure is a Cartan--Hadamard manifold, or more generally: a reduced Birkhoff--Beatley system. Some exotic behavior of convex hulls are also studied.

In this note, we give a short solution of the kissing number problem in dimension three.

A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. The aim of this paper is to investigate three questions of Conway, regarding monostable polyhedra, which first appeared in a 1969 paper of Goldberg and Guy (M. Goldberg and R.K. Guy, Stability of polyhedra (J.H. Conway and R.K. Guy), SIAM Rev. 11 (1969), 78-82). In this note we answer two of these problems and make a conjecture about the third one. The main tool of our proof is a general theorem describing approximations of smooth convex bodies by convex polyhedra in terms of their static equilibrium points. As another application of this theorem, we prove the existence of a convex polyhedron with only one stable and one unstable point.

We show that the fifth and the eighth Busemann-Petty problems have positive solutions for bodies that are sufficiently close to the Euclidean ball in the Banach-Mazur distance.

We develop the theory of APD profiles introduced by J. Dydak for ∞ -pseudometric spaces. We connect them with transfinite asymptotic dimension defined by T. Radul. We give a characterization of spaces with transfinite asymptotic dimension at most ω+n for n∈ω and a sufficient condition for a space to have transfinite asymptotic dimension at most m⋅ω+n for m,n∈ω , using the language of APD profiles.