Is there any test to tell me whether a straight line in a 3D euclidean space passes through a bounded closed convex region? To focus on a more specialised version of the problem, y …

Let $x_1,\dots,x_n$ be a set of points in $\mathbb{R}^3$ and let $\mathcal{O}_1 ,\dots, \mathcal{O}_m$ denote a set of polyhedral obstacles. What is the name for the surfaces that …

I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups) 1) in mathematics 2) outside of mathematics, besides …

Let $Q$ be a smooth manifold, and let $G$ be a Lie group which acts smoothly on $Q$ on the left. Question 1: does the group $G$ act naturally on the tangent bundle $TQ \to Q$? …

This is a cross-posting of a MSE question (which did not receive any feedback there so far). Say that a vector $x=(x_1,x_2, \ldots ,x_n)\in {\mathbb R}^n$ is nondecreasing if $x_1 …

I am looking for an attractive, but rigorous definition of area; say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built …

Let $\alpha:\mathbb{R}^n\to\mathbb{R}^n$, $n\geq 2$, be a $\mathbb{Q}$-linear bijection with the following properties: 1) $\alpha$ sends straight affine $\mathbb{R}$-lines to stra …

Let $x_1,\ldots,x_k \in \mathbb{R}^d$ be $k$ unit vectors in $d$ dimensional Euclidean space, and let $S = \mathrm{span}(x_1,\ldots,x_k)$ be a linear subspace defined by these poin …

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists …

In three-dimensional euclidean space, consider the closed unit ball $B$. Let $T$ be a tetrahedron, and $E$ an ellipse, with $E \subset T \subset B$. Does there necessarily exist a …

Let $F \subset \mathbb R$ be the set of lengths of line segments that one can construct, starting from the points $(0,0)$ and $(1,0)$, using a straightedge, compass, and an Archime …

Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof. Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c …

Let us develop the question: Let us focus on finite real vector space, equiped with a norm. A priori, one does not make the hypothesis that the norm is derived from a scalar prod …

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and sh …

I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here. In the Wikipedia page on …