Mathematics

We shall prove a convergence result relative to sequences of Minkowski symmetrals of general compact sets. In particular, we investigate the case when this process is induced by sequences of subspaces whose elements belong to a finite family, following the path marked by Klain in [13], and the generalizations in [4] and [2]. We prove an analogue result for Fiber symmetrization of a specific class of compact sets. The idempotency for symmetrization of this family of sets is investigated, leading to a simple generalization of a result from Klartag [14] regarding the approximation of a ball through a finite number of symmetrizations, and generalizing an approximation result in [9]

We study a generalization of the Bakry-Émery pointwise gradient estimate for the heat semigroup and its equivalence with some entropic inequalities along the heat flow and Wasserstein geodesics for metric-measure spaces with a suitable group structure. Our main result applies to Carnot groups of any step and to the SU(2) group.

We present a result which simultaneously extends the Busemann intersection inequality to the case of non-integer moments of the corresponding volumes and the Busemann random simplex inequality to the case of simplices of smaller dimensions.

Monsky's celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation f among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial p , also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monsky's polynomial f can be chosen to be invariant under deformation. Although f is not uniquely defined, the interplay between p and f then allows us to identify a canonical pair of choices for the polynomial f . In many cases, all of the coefficients of the canonical f polynomials are positive. We also use the deformation-invariance of f to prove that the polynomial p is congruent modulo 2 to a power of the sum of its variables.

Motivated by classical Euler's Tonnetz , we introduce and study the combinatorics and topology of more general simplicial complexes Ton n n,k (L) of "Tonnetz type". Out main result is that for a sufficiently generic choice of parameters the generalized tonnetz Ton n n,k (L) is a triangulation of a (k−1) -dimensional torus T k−1 . In the proof we construct and use the properties of a "discrete Abel-Jacobi map", which takes values in the torus T k−1 ≅ R k−1 /Λ where Λ≅ A ∗ k−1 is the permutohedral lattice.

Chessboard complexes and their generalizations, as objects, and Discrete Morse theory, as a tool, are presented as a unifying theme linking different areas of geometry, topology, algebra and combinatorics. Edmonds and Fulkerson bottleneck (minmax) theorem is proved and interpreted as a result about a critical point of a discrete Morse function on the Bier sphere of an associated simplicial complex K . We illustrate the use of "standard discrete Morse functions" on generalized chessboard complexes by proving a connectivity result for chessboard complexes with multiplicities. Applications include new Tverberg-Van Kampen-Flores type results for j -wise disjoint partitions of a simplex.

We study generalizations of Lorentzian warped products with one-dimensional base of the form I × f X , where I is an interval, X is a length space and f is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [Kunzinger, Sämann; Ann. Glob. Anal. Geom. 54(3):399--447, 2018], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber X . The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.

We define the notion of near geodesic between points of a metric space when no geodesic exists, and use this to extend Recio-Mitter's notion of geodesic complexity to non-geodesic spaces. This has potential application to topological robotics. We determine explicit near geodesics and geodesic complexity in a variety of cases.

We show that every geodesic metric space admitting an injective continuous map into the plane as well as every planar graph has Nagata dimension at most two, hence asymptotic dimension at most two. This relies on and answers a question in a very recent work by Fujiwara and Papasoglu. We conclude that all three-dimensional Hadamard manifolds have Nagata dimension three. As a consequence, all such manifolds are absolute Lipschitz retracts.

Horospherical products of two hyperbolic spaces unify the construction of metric spaces such as the Diestel-Leader graphs, the SOL geometry or the treebolic spaces. Given two proper, geodesically complete, Gromov hyperbolic, Busemann spaces H p and H q , we study the geometry of their horospherical product H:= H p ⋈ H q through a description of its geodesics. Specifically we introduce a large family of distances on H p ⋈ H q . We show that all these distances produce the same large scale geometry. This description allows us to depict the shape of geodesic segments and geodesic lines. The understanding of the geodesics' behaviour leads us to the characterization of the visual boundary of the horospherical products. Our results are based on metric estimates on paths avoiding horospheres in a Gromov hyperbolic space.