Mathematics

We provide a Rademacher theorem for intrinsically Lipschitz functions ϕ:U⊆W→L , where U is a Borel set, W and L are complementary subgroups of a Carnot group, where we require that L is a normal subgroup. Our hypotheses are satisfied for example when W is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions.

A new intrinsic metric called t -metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains G⊊ R n . The behaviour of the new metric is also studied under a few examples of conformal and quasiconformal mappings, and the differences between the balls drawn with all the metrics considered are compared by both graphical and analytical means.

We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their triangle centers. We show that (i) for any concentric pair, the power of the center with respect to either circumcircle or Euler's circle is invariant, and (ii) if a triangle center of a 3-periodic in a generic nested pair is a fixed affine combination of barycenter and circumcenter, its locus over the family is an ellipse.

We study self-intersected N-periodics in the elliptic billiard, describing new facts about their geometry (e.g., self-intersected 4-periodics have vertices concyclic with the foci). We also check if some invariants listed in "Eighty New Invariants of N-Periodics in the Elliptic Billiard" (2020), arXiv:2004.12497, remain invariant in the self-intersected case. Toward that end, we derive explicit expressions for many low-N simple and self-intersected cases. We identify two special cases (one simple, one self-intersected) where a quantity prescribed to be invariant is actually variable.

We have shown recently that, given a metric space X , the coarse equivalence classes of metrics on the two copies of X form an inverse semigroup M(X) . Here we study the property of idempotents in M(X) of being finite or infinite, which is similar to this property for projections in C*-algebras. We show that if X is a free group then the unit of M(X) is infinite, while if X is a free abelian group then it is finite. As a by-product, we show that the inverse semigroup M(X) is not a quasi-isometry invariant. More examples of finite and infinite idempotents are provided. We also give a geometric description of spaces, for which their inverse semigroup M(X) is commutative.

We say that a triangle T tiles a polygon A , if A can be dissected into finitely many nonoverlapping triangles similar to T . We show that if N>42 , then there are at most three nonsimilar triangles T such that the angles of T are rational multiples of π and T tiles the regular N -gon. A tiling into similar triangles is called regular, if the pieces have two angles, $\al$ and $\be$, such that at each vertex of the tiling the number of angles $\al$ is the same as that of $\be$. Otherwise the tiling is irregular. It is known that for every regular polygon A there are infinitely many triangles that tile A regularly. We show that if N>10 , then a triangle T tiles the regular N -gon irregularly only if the angles of T are rational multiples of π . Therefore, the numbers of triangles tiling the regular N -gon irregularly is at most three for every N>42 .

Iso-edge domains are a variant of the iso-Delaunay decomposition introduced by Voronoi. They were introduced by Baranovskii & Ryshkov in order to solve the covering problem in dimension 5 . In this work we revisit this decomposition and prove the following new results: ??We review the existing theory and give a general mass-formula for the iso-edge domains. ??We prove that the associated toroidal compactification of the moduli space of principally polarized abelian varieties is projective. ??We prove the Conway--Sloane conjecture in dimension 5 . ??We prove that the quadratic forms for which the conorms are non-negative are exactly the matroidal ones in dimension 5 .

We investigate isometric embeddings of finite metric trees into ( R n , d 1 ) and ( R n , d ∞ ) . We prove that a finite metric tree can be isometrically embedded into ( R n , d 1 ) if and only if the number of its leaves is at most 2n . We show that a finite star tree with at most 2 n leaves can be isometrically embedded into ( R n , d ∞ ) and a finite metric tree with more than 2 n leaves cannot be isometrically embedded into ( R n , d ∞ ) . We conjecture that an arbitrary finite metric tree with at most 2 n leaves can be isometrically embedded into ( R n , d ∞ ) .

Let (X,d,m) be a metric measure space. The study of the Wasserstein space ( P p (X), W p ) associated to X has proved useful in describing several geometrical properties of X. In this paper we focus on the study of isometries of P p (X) for p??1,?? under the assumption that there is some characterization of optimal maps between measures, the so called Good transport behaviour GT B p . Our first result states that the set of Dirac deltas is invariant under isometries of the Wasserstein space. Additionally we obtain that the isometry groups of the base Riemannian manifold M coincides with the one of the Wasserstein space P p (M) under assumptions on the manifold; namely, for p=2 that the sectional curvature is strictly positive and for general p??1,?? that M is a Compact Rank One Symmetric Space.

We show that for any compact convex set K in R d and any finite family F of convex sets in R d , if the intersection of every sufficiently small subfamily of F contains an isometric copy of K of volume 1 , then the intersection of the whole family contains an isometric copy of K scaled by a factor of (1−ε) , where ε is positive and fixed in advance. Unless K is very similar to a disk, the shrinking factor is unavoidable. We prove similar results for affine copies of K . We show how our results imply the existence of randomized algorithms that approximate the largest copy of K that fits inside a given polytope P whose expected runtime is linear on the number of facets of P .