Related papers: On the Erdos-Mordell Inequality for Triangles in T…
We resolve a conjecture of De Palma and Trevisan by proving the triangle inequality for a quantum 2-Wasserstein distance. The proof relies on complex analysis methods to establish a new integral representation of the cost in the optimal…
A celebrated result in convex geometry is Gr\"unbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Gr\"unbaum-type…
The author proposes a new geometry in this book. The author named this new geometry Intercenter Geometry. Intercenter Geometry is different from traditional Euclidean geometry and analytic geometry (coordinate geometry). The idea of…
In this paper, we prove that a natural candidate for a homogeneous norm on a graded Lie algebra of any length satisfies the triangle inequality which answers Moskowitz's question.
We establish a sharp inequality between the blocks of positive partitioned matrices and conjecture a triangle type inequality for contractions: Given three contactions A,B,C, we conjecture that the constant c=3/4 is sharp in the triangle…
In this article, we make use of some known method to investigate some properties of the numbers represented as sums of two equal odd powers, i.e., the equation $x^n+y^n=N$ for $n\ge3$. It was originated in developing algorithms to search…
We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…
While the concept of straight-line length is well understood in taxicab geometry, little research has been done into the length of curves or the nature of area and volume in this geometry. This paper sets forth a comprehensive view of the…
A convex duality result for martingale optimal transport problems with two marginals was established in Beiglb\"ock et al. (2013). In this paper we provide a generalization of this result to the multi-period setting.
Distances are pervasive in machine learning. They serve as similarity measures, loss functions, and learning targets; it is said that a good distance measure solves a task. When defining distances, the triangle inequality has proven to be a…
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 obtain an analytical solution for the weighted Fermat-Torricelli problem for an equilateral geodesic triangle A_1A_2A_3 which is composed by three equal geodesic arcs (sides) of length Pi/2 for given three positive unequal weights that…
In this article we determine the class of triangles $A_iB_iC_i$ which are orthohomological with a given triangle $ABC$ and inscribed in the triangle $ABC$ (with $A_i \in BC$, $B_i \in CA$ and $C_i \in AB$).
For positive definite matrices $A$ and $B$, the Araki-Lieb-Thirring inequality amounts to an eigenvalue log-submajorisation relation for fractional powers $$\lambda(A^t B^t) \prec_{w(\log)} \lambda^t(AB), \quad 0<t\le 1,$$ while for…
We present a complete system of inequalities for the inradius, circumradius, and diameter in the $3$-dimensional Euclidean space. To do so, we prove quasiconcavity of the inradius evaluated over $n$-simplices with a common facet…
Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the vertices of the triangulation. Moreover, i -…
We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane, such as: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude,…
In this short note the authors give answers to the three open problems formulated by Wu and Srivastava [{\it Appl. Math. Lett. 25 (2012), 1347--1353}]. We disprove the Problem 1, by showing that there exists a triangle which does not…
Some trapezoid and mid-point type inequalities related to the Hermite-Hadamard inequality for the mappings defined on a ball in the space are obtained.
We discuss transportation cost inequalities for uniform measures on convex bodies, and connections with other geometric and functional inequalities. In particular, we show how transportation inequalities can be applied to the slicing…