Related papers: The Euler and Grace-Danielsson inequalities for ne…
The difference between the (squared) sides of the Grace-Danielsson inequality for tetrahedra will be represented as a sum of two nonnegative terms. This gives another proof of the inequality. Examining the denominator allows us to…
Euler's inequality is a well known inequality relating the inradius and circumradius of a triangle. In Euclidean geometry, this inequality takes the form $R \geq 2r$ where $R$ is the circumradius and $r$ is the inradius. In spherical…
We propose 3D generalizations of the Feuerbach theorem: the first one deals with a tetrahedron analogue of the Euler circle, the second one is done by means of an {\guillemotleft}up-in-ex-touch{\guillemotright} construction. Then we give a…
Given a Euclidean simplex of dimension $n\geqslant 2$ let its radii of inscribed and circumscribed spheres be $r$ and $R$, and the distance between the centers of the inscribed and circumscribed spheres be $d.$ Then, $(R-nr)(R+(n-2)r)…
We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region…
The determination of the gravitational potential by the polyhedral method is revisited in the case where the surface of a body is composed of triangular facets. Based upon six test-shapes of astrophysical interest (sphere, spheroid,…
Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use…
We prove that if $E \subset {\mathbb F}_q^2$, $q \equiv 3 \mod 4$, has size greater than $Cq^{7/4}$, then $E$ determines a positive proportion of all congruence classes of triangles in ${\mathbb F}_q^2$. The approach in this paper is based…
We give a sufficient condition for branched minimal immersions of spheres into ellipsoids to be embedded: we show that if the coordinate functions of the branched minimal immersion are first or second eigenfunctions with respect to a…
Embedding graphs in a geographical or latent space, i.e.\ inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We…
We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the…
We solve a very classical problem motivated by considerations in quantum gravity: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms…
This article is covered by the article arxiv.1012.0925 We study intersection of two polyhedral spheres without self-intersections in 3-space. We find necessary and sufficient conditions on sequences x = x_1,x_2,...,x_n, y = y_1,y_2,...,y_n…
First, we fill in key gaps in Steiner's nice characterization of the most nearly circular ellipse which passes through the vertices of a convex quadrilateral, D. Steiner proved that there is only one pair of conjugate directions, M1 and M2,…
In non-Euclidean geometry, there are several known correspondings to Chapple-Euler Theorem. This remark shows that those results yield expressions corredponding to the well-known formula $d=\sqrt{R(R-2r)}$.
We construct a counterexample for an analogue of Masur's criterion in the setting of Teichm\"uller space equipped with the Thurston metric. For that, we find a minimal, filling, non-uniquely ergodic lamination $\lambda$ on the seven-times…
A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance…
Certain triangle inequalities involving the circumradius, inradius, and side lengths of a triangle are generalized to spherical and hyperbolic geometry. Examples include strengthenings of Euler's inequality, $R\geq2r$. An extension of…
We investigate the emergence of finite-amplitude non-zonal flows on the sphere $\mathbb{S}^2$ arising from stationary solutions to the 2D Euler equations. By restricting the Laplace-Beltrami eigenspace to the invariant subspace of the…
We geometrically prove that in a d-dimensional cube with edges of length n, the number of particular d-dimensional tetrahedrons are given by Eulerian numbers. These tetrahedrons tassellate the cube, In this way the sum of the cubes are the…