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There was quite a bit of work on the so-called equichordal problem throughout the 20th century, to decide if some plane convex curve could have two equichordal points. A point is equichordal for a clo …

Suppose one intersects unit-radius solid cylinders in $\mathbb{R}^3$, with each cylinder axis passing through the origin. For example, two such cylinders produce the Steinmetz solid. But if we imagin …

This is driven more by curiosity than by research, but nevertheless may be of some interest. Start with a regular tetrahedron $T$ with corners $(a,b,c,d)$, and let $x$ be its incenter—the center of t …

Dominic van der Zypen posed an interesting Box stacking problem. This is a spin-off question. Let a collection of rectangles $r_1,\ldots,r_n$ be given by their side lengths in $\mathbb{R}$. Let $R$ b …

I heard it claimed that there is, in some sense, only one random metric on $\mathbb{S}^2$. I would appreciate any pointer to literature that explicates this intriguing claim. So far my own searches ha …

This is a variation on an earlier question resolved by user35353: Can a tangle of arcs interlock? In that question, the arcs were restricted to circular arcs, and user35353's proof that one arc can be …

What is the limit, as $n \to \infty$, of the expected distance between two points chosen uniformly at random within a unit edge-length hypercube in $\mathbb{R}^n$? For $n=1$, the average dist …

This is a companion contrast to the earlier analogous question for unit $n$-cubes, where the answer (provided by several respondents) is $\infty$ . What is the limit, as $n \to \infty$, of the exp …

It was established by TMA, @WillSawin, and @DouglasZare, in their responses to the MO question, "Thales' semicircle theorem in higher dimensions," that the natural generalization of Thales' semicircle …

The volume of an $n$-dimensional simplex of unit edge length is $$V(n) = \frac{\sqrt{n+1}}{n! 2^{n/2}} \;,$$ so at least $\lceil 1/V(n) \rceil$ such simplices are needed to cover the unit $n$-cube. …

I have two 3D rotations about the origin, represented as $3 \times 3$ orthogonal matrices $M_1$ and $M_2$ (specified by numerical entries), and I would like to interpolate (and compute) a continuous s …

This is a followup to Bill Thurston's question about different notions of hulls. Here I want to raise a question about the reflex-free hull, which is, intuitively, the smallest enclosing shape to an o …

Define a circle as a geometric circle of positive, finite radius: a set of points in $\mathbb{R}^3$ congruent to the set $x^2 + y^2 = r^2$ in the $xy$-plane. [Edited as per BMann's comment.] I am inte …

I am wondering what can be inferred when a discrete gradient ascent algorithm gets stuck in a cycle. Here is the situation. A function $f(x,y)$ is defined over a range $[0,n]^2$, and the algorithm wal …

Let $S$ be the surface of a compact, convex, smooth ($C^\infty$) body in $\mathbb{R}^3$, with strictly positive Gaussian curvature at every point of $S$. Fix a direction $z$ in a Cartesian coordinate …