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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: $bullet$ We review the existing theory and give a general mass-formula for the iso-edge domains. $bullet$ We prove that the associated toroidal compactification of the moduli space of principally polarized abelian varieties is projective. $bullet$ We prove the Conway--Sloane conjecture in dimension $5$. $bullet$ We prove that the quadratic forms for which the conorms are non-negative are exactly the matroidal ones in dimension $5$.
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The Mobius metric $delta_G$ is studied in the cases where its domain $G$ is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the Mobius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the Mobius metric and its connection to the hyperbolic metric in polygon domains.
We consider an optimal stretching problem for strictly convex domains in $mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant $1$. Specifically, we prove that the stretched convex domain which captures the most positive lattice points in the large volume limit is balanced: the $(d-1)$-dimensional measures of the intersections of the domain with each coordinate hyperplane are equal. Our results extend those of Antunes & Freitas, van den Berg, Bucur & Gittins, Ariturk & Laugesen, van den Berg & Gittins, and Gittins & Larson. The approach is motivated by the Fourier analysis techniques used to prove the classical $#{(i,j) in mathbb{Z}^2 : i^2 +j^2 le r^2 } =pi r^2 + mathcal{O}(r^{2/3})$ result for the Gauss circle problem.
Suppose that $E$ and $E$ denote real Banach spaces with dimension at least 2, that $Dsubset E$ and $Dsubset E$ are domains, and that $f: Dto D$ is a homeomorphism. In this paper, we prove the following subinvariance property for the class of uniform domains: Suppose that $f$ is a freely quasiconformal mapping and that $D$ is uniform. Then the image $f(D_1)$ of every uniform subdomain $D_1$ in $D$ under $f$ is still uniform. This result answers an open problem of Vaisala in the affirmative.
We prove in the setting of $Q$--Ahlfors regular PI--spaces the following result: if a domain has uniformly large boundary when measured with respect to the $s$--dimensional Hausdorff content, then its visible boundary has large $t$--dimensional Hausdorff content for every $0<t<sleq Q-1$. The visible boundary is the set of points that can be reached by a John curve from a fixed point $z_{0}in Omega$. This generalizes recent results by Koskela-Nandi-Nicolau (from $mathbb{R}^2$) and Azzam ($mathbb{R}^n$). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.
Monskys 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 Monskys 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.
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