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Maybe not an exact answer, but some thoughts to get a foot in the door: no three points must be collinear, because otherwise a triangle with area $0$ would exist and thus every set of points in ge …

A more practical heuristic for generating dispersed point sets inside a compact region $\Omega$, that also generates to higher dimensions is the following, which I would like to call the stay away fr …

Let $\Delta := (A,B,C)$ be a triangle that is defined by three points in the Euclidean plane that are not collinear. Let further $E_{(A,B),\,C},\,E_{(C,A),\,B},\,E_{(B,A),\,B}$ be the set of ellipses …

if you look for non-circular gear you at least find the wiki article that also addresses the mathematics; there are also links to publication on the subject.

Let $H,H'\subset\mathbb{R}^m$ be two hyperplanes with unit normal-vector, and let $P\subset\mathbb{R^m}$ be a convex polytope (defined via its corners $v_0, ... , v_n$, where $n\ge m$). Let's furth …

It just occured to me that the question can be answered in the affirmative way for the following reasons: the order relation doesn't change if the distance sum is divided by some positive constant …

As the envelope is made of paper, a mathematical model of its deformation would be an isometric coordinate transformation and thus must have zero Gauss curvature almost everywhere; that restriction is …

This question is motivated by an almost successful attempt to define planar convex hulls of a finite set of isolated points only via subset sums of the distances between point-pairs; the advantage of …

By "Pair Statistics" I understand statics that are based on values $\varphi:\mathcal{P}\times\mathcal{P}\ni(p,q)\mapsto y\in\mathbb{R}$ that can be observed for every pair $(p,q)$ of individuals of a …

If $P$ is a set of $n$ points in the euclidean plane whose convex hull $\operatorname{CH}(P)$ has $h$ corners, and $Q\subset P$ has $m\le\lfloor\frac{h}{2}\rfloor$ points and maximal sum of pairwise d …

Let $A,B,C,D$ be the corners of a tetrahedron with positive volume and distinct sidelengths. Is there a positive $x$ and a planar straight-line embedding of a $K_4$ graph with distinct vertices $A’,B’ …

An algorithm that works at least for dimensions $2$ and $3$ is: calculate a spanning tree of the $n$ points calculate the bisector planes of the spanning tree's edges the sought poiint is in the inte …

A simple "a posteriori" criterion is that on the optimal tour the distances to the tour-neighbors is smaller than that to any of the other vertices. Convexity alone doesn't suffice as the example of e …

to me the best possible solution seems to be the sequence of closed Hilbert curves on the Cubed Sphere; that curve consists of six ordinary Hilbert Curves, one for each face of a cube, which, when app …

This question could be seen as a coordinate-free variant of Sylvester's Four Point Problem (cf e.g. http://mathworld.wolfram.com/SylvestersFour-PointProblem.html): Suppose one are given an inifinit …