Additive Reverses of the Generalised Triangle Inequality in Normed Spaces

非欧几里得平面中的广义刚体运动及其在全局刚性中的应用

Generalised rigid body motions in non-Euclidean planes with applications to global rigidity

针对一般实赋范平面中杆-铰框架的全局刚性问题，本文引入由解析范数决定的广义刚体运动理论，并据此得出若干几何与组合结论。

Coincident-point rigidity in normed planes

A bar-joint framework $(G,p)$ is the combination of a graph $G$ and a map $p$ assigning positions, in some space, to the vertices of $G$. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a s…

Monge points, Euler lines, and Feuerbach spheres in Minkowski spaces

It is surprising, but an established fact that the field of Elementary Geometry referring to normed spaces (= Minkowski spaces) is not a systematically developed discipline. There are many natural notions and problems of elementary and classical geometry that were never investigated in this more general framework, although their Euclidean subcases are well known and this extended viewpoint is promising. An example is the geometry of simplices in non-Euclidean normed spaces; not many papers in this direction exist. Inspired by this lack of natural results on Minkowskian simplices, we present a …