度量几何
We pursue the investigation of the concept of $n$-distance, an $n$-variable version of the classical concept of distance recently introduced and investigated by Kiss, Marichal, and Teheux. We especially focus on the challenging problem of…
For Riemannian metrics of constant positive curvature on a punctured sphere with conic singularities at the punctures and co-axial monodromy of the developing map, possible angles at the singularities are completely described. This…
This article extends the range of 1-DOF rigid-foldable developable quadrilateral creased papers. In a previous article, we put forward a sufficient and necessary condition for a quadrilateral creased paper to be rigid-foldable, and…
We develop some design examples for approximating a target surface at the final rigidly folded state of a developable quadrilateral creased paper, which is folded with a 1-DOF rigid folding motion from the planar state. The final rigidly…
All continuous translation invariant complex-valued valuations on Lebesgue measurable functions are completely classified. And all continuous rotation invariant complex-valued valuations on spherical Lebesgue measurable functions are also…
The Colin de Verdiere number of graph G, denoted by \mu(G), is a spectral invariant of G that is related to some of its topological properties. For example, \mu(G) \leq 3 iff G is planar. A penny graph is the contact graph of equal-radii…
A quasiconformal tree is a metric tree that is doubling and of bounded turning. We prove that every quasiconformal tree is quasisymmetrically equivalent to a geodesic tree with Hausdorff dimension arbitrarily close to 1.
We investigate the uniqueness for the Monge-Amp\`{e}re type equation \begin{equation} \label{eq-abstract} det(u_{ij}+\delta_{ij}u)_{i,j=1}^{n-1}=G(u),\ \ \ \ \ \ \ (*)\end{equation}on $S^{n-1}$, where $u$ is the restriction of the support…
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…
Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are…
We study metric spaces defined via a conformal weight, or more generally a measurable Finsler structure, on a domain $\Omega \subset \mathbb{R}^2$ that vanishes on a compact set $E \subset \Omega$ and satisfies mild assumptions. Our main…
We derive fundamentally new equations that are satisfied by first-order flexes of a flexible polyhedron. Moreover, we indicate two sources of such new equations. These sources are the Dehn invariants and rigidity matrix. The equations…
We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed…
Being motivated by the theory of flexible polyhedra, we study the Dirichlet and Neumann eigenvalues for the Laplace operator in special bounded domains of Euclidean $d$-space. The boundary of such a domain is an embedded simplicial complex…
In this note we show that locally $p$-admissible measures on $\mathbb{R}$ necessarily come from local Muckenhoupt $A_p$ weights. In the proof we employ the corresponding characterization of global $p$-admissible measures on $\mathbb{R}$ in…
Taking up a suggestion of David Gale from 1956, we generate sets of combinatorially isomorphic polytopes by choosing their Gale diagrams at random. We find that in high dimensions, and under suitable assumptions on the growth of the…
In this note, we investigate an infinite one parameter family of circle packings, each with a set of three mutually tangent circles. We use these to generate an infinite set of circle packings with the Apollonian property. That is, every…
The complex conjecture of Stefan Banach states that if V is a Banach space over the complex numbers where for some n, 1<n<dim(V), all of its n-dimensional subspaces are isometric, then V is a Hilbert space. Mikhail Gromov proved it for n…
In this paper, we introduce quasi-convex subsets in Alxandrov spaces with lower curvature bound, which include not only all closed convex subsets without boundary but also all extremal subsets. Moreover, we explore several essential…
We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $\Gamma$-permutahedron for some finite reflection group $\Gamma\subset\mathrm{O}(\mathbb R^d)$. The same holds true for zonotopes…