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consider the set of lines defined by all pairs of points $\lbrace[(u,v),(u-v,v+u)],\ u,v\in\mathbb{N}\rbrace$ in the euclidean plane Question: what is kown about the density of the set of intersectio …

Given a finite set $P$ of $n\gt d+1$ points in $d$-dimensional euclidean space. Under the assumption that the points of $P$ are in general position in the sense that $\lbrace p_{i_1},\dots,p_{i_d}\rbr …

The triangles of a planar Delaunay Triangulations are essentially characterized by the property that no triangle's corner is inside another triangle's circumcircle; Delaunay Triangulations can be calc …

Question: given the triangulation $T$ of a set $P$ of $n$ points $p_1,\dots,p_n$ in the euclidean plane whose convex hull is a triangle, can we always find a set $Q$ of $n+1$ points $q_0,q_1,\dots,q_n …

Question: what is known about the sequence $\mathbb{X}\subset \mathbb{N}_0$ such that for each $k\in \mathbb{X}$ there exists a set of $n$ points in general position in the Euclidean plane such that t …

Let $P$ be a closed polygon defined by the sequence $p_0,\,\dots,\,p_{n-1},p_0$ of points. Question: how can one construct, with straightedge and compass alone, another sequence of points $q_0,\,\dot …

Question: What is known about the problem of covering a finite set of $\mathbb{P}$ of points in the plane with convex polygons that have the same number $m$ of points from $\mathbb{P}$ as corners and …

It is known that the Minimum Spanning Tree (MST) of a finite set of points in the Euclidean plane is contained in the point set's Delaunay triangulation, but is that all that can be said about their r …

You must be careful with what you are actually asking for; the criteria you give as examples are valid for every simple polygon and do not characterize simple polygons of shortest perimeter with a giv …

I call a spanning tree perfectly balanced if, after a two-coloring of the tree-graph's vertices the two vertex sets that are defined by the assigned colors have equal cardinality and the two vertex s …

This should rather be a comment, posting it as an answer is to give a visual clue that there is something in the vein of a proof. Elaborating on Timothy Chow's insightful comment it suffices to consid …

A complete symmetric graph with $n=4$ vertices, i.e. a $K_4$ is the disjoint union of three perfect matchings $M_{\text{min}},M_{\text{mid}},M_{\text{max}}$ of which $M_{\text{min}}$ denotes the light …

Let $T_{\mathrm{MIN}}$ and $T_{\mathrm{MAX}}$ denote the shortest, resp. longest Hamilton cycle through a set of $n=2k+1$ points. Let further $S_{\mathrm{MIN}}$ and $S_{\mathrm{MAX}}$ be the "antipoda …

Question: is there an established name for sets of $n$ points in the euclidean plane whose shortest Hamilton cycles consists of the $n$ pairs of points having the $n$ smallest distances? Names for sym …

That should actually be a seen as comment: It appears as an $O(n^2)$ is possible: suppose $q_{ij}, q_{ik}$ are given and $\varphi(q_{ij})=0^\circ$ and $0^\circ\lt\varphi(q_{ik})\le 180^\circ$ then the …