Mathematics

In this work we study the set of strictly accretive matrices, that is, the set of matrices with positive definite Hermitian part, and show that the set can be interpreted as a smooth manifold. Using the recently proposed symmetric polar decomposition for sectorial matrices, we show that this manifold is diffeomorphic to a direct product of the manifold of (Hermitian) positive definite matrices and the manifold of strictly accretive unitary matrices. Utilizing this decomposition, we introduce a family of Finsler metrics on the manifold and charaterize their geodesics and geodesic distance. Finally, we apply the geodesic distance to a matrix approximation problem, and also give some comments on the relation between the introduced geometry and the geometric mean for strictly accretive matrices as defined by S. Drury in [S. Drury, Linear Multilinear Algebra. 2015 63(2):296-301].

We prove that the L 4 norm of the vertical perimeter of any measurable subset of the 3 -dimensional Heisenberg group H is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the L 4 norm replaced by the L q norm for any q<4 . This is in contrast to the 5 -dimensional setting, where the above result holds with the L 4 norm replaced by the L 2 norm. The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in H . In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the L 1 distortion of a word-ball of radius n≥2 in the discrete 3 -dimensional Heisenberg group is bounded above and below by universal constant multiples of logn − − − − √ 4 ; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius n≥2 is of order logn − − − − √ .

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension d= 2 2l+1 over the field F 3 2 . We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.

In this note we introduce the problem of illumination of convex bodies in spherical spaces and solve it for a large subfamily of convex bodies. We derive from it a combinatorial version of the classical illumination problem for convex bodies in Euclidean spaces as well as a solution to that for a large subfamily of convex bodies, which in dimension three leads to special Koebe polyhedra.

In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda. Furthermore, we study the necessity of conditions on the underlying space in Federer's characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.

We say that the sequence of groups Γ n acts almost transitively on a sequence of proper metric spaces X n if for every n , there is an isometric discrete cocompact action of Γ n on X n such that the diameters of the quotients X n / Γ n converge to 0 as n→∞ . In such a case, we prove that if the sequence X n consists of length spaces and converges in the pointed Gromov--Hausdorff sense to a proper length space X , then X is isometric to a nilpotent locally compact group equipped with an invariant length metric. Furthermore, assuming X is either finite dimensional or semilocally simply connected, we show that it is a Lie group equipped with a Finsler or sub-Finsler metric, and for large enough n , there are subgroups Λ n ≤ π 1 ( X n ) with surjective morphisms Λ n → π 1 (X) .

We prove a ? -convergence result for Cheeger energies along sequences of metric measure spaces, where the measure space is kept fixed, while distances are monotonically converging from below to the limit one. As a consequence, we show that the infinitesimal Hilbertianity condition is stable under this kind of convergence of metric measure spaces.

We introduce a family of dimensions, which we call the Φ -intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. We do this by restricting the allowable covers in the definition of Hausdorff dimension, but in a wider variety of ways than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We investigate relationships between the Φ -intermediate dimensions and other notions of dimension, and study many analytic and geometric properties of the dimensions. We prove continuity-like results which improve similar results for the intermediate dimensions and give a sharp general lower bound for the intermediate dimensions that is positive for all θ∈(0,1] for sets with positive box dimension. We prove Hölder distortion estimates which imply bi-Lipschitz stability for the Φ -intermediate dimensions. We prove a mass distribution principle and Frostman type lemma, and use these to study dimensions of product sets, and to show that the lower versions of the dimensions, unlike the the upper versions, are not finitely stable. We show that for any compact subset of an appropriate space, these dimensions can be used to `recover the interpolation' between the Hausdorff and box dimensions of sets for which the intermediate dimensions are discontinuous at θ=0 , thus providing more refined geometric information about such sets.

We study the general rational trigonometry of a tetrahedron, based on quadrances, spreads and solid spreads, using vector products associated to an arbitrary symmetric bilinear form over a general field, not of characteristic two. This gives us algebraic analogs of many classical formulas, as well as new insights and results. In particular we derive original relations for a tri-rectangular tetrahedron.

In this article we introduce a general definition of the concept of center of an n -gon, for n≥3 , generalizing the idea of C. Kimberling for triangle. We define centers associated to functions instead of to geometrical properties. We discuss the definition of those functions depending on both, the vertices of the polygons or the lengths of sides and diagonals. We explore the problem of characterization of regular polygons in terms of these n -gon center functions and we study the relation between our general definition of center of a polygon and other approaches arising from Applied Mathematics.