# Tag Info

Accepted

### Shortest closed curve to inspect a sphere

James Wenk and I just finished a paper proving Zalgaller's sphere inspection conjecture for closed curves: Shortest closed curve to inspect a sphere. We show that in $R^3$ any closed curve $\gamma$ ...
• 7,057
Accepted

### Which polygons can be turned inside out by a smooth deformation?

This question was explored here: Lenhart, William J., and Sue H. Whitesides. "Reconfiguring closed polygonal chains in Euclidean $d$-space." Discrete & Computational Geometry 13, no. 1 (1995): ...
• 149k
Accepted

• 104k

### Shortest path connecting two opposite points on a cube

Take a path that joins the antipodes and concatenate it with its symmetric image. Get a centrally symmetric closed path on the boundary of the cube. If this path avoids one of the facets of the cube (...
• 6,277
Accepted

### Furthest distance half the diameter?

Denote the diameter by $d$ and distance by $|x-y|$. Then there are $y,z$ such that $d=|y-z|$ and we have by triangle inequality for every $x$: $$d=|y-z|\leq |y-x|+|x-z|\leq 2f(x),$$ so we obtain your ...
• 89.7k
Accepted

### Convex set with no interior contained in hyperplane?

Based on Jack's comment. The "Hilbert cube" in Hilbert space $l^2$. $$C :=\{(x_1,x_2,\dots) : |x_k| \le 2^{-k}\;\forall k\}$$ $C$ is convex, compact (so it has empty interior) but has dense ...
• 40.6k
Accepted

### Covering the unit sphere in $\mathbf{R}^n$ with $2n$ congruent disks

For $n=3$ the answer is yes, as was shown by Fejes Tóth in 1943; see the Theorem on p.34 of his book Regular Figures. For $n=4$ the answer is also positive as shown in the 2000 paper, The blocking ...
• 7,057

### Shortest closed curve to inspect a sphere

The baseball stitches curve suggested by Gjergji Zaimi appears in another paper of Zalgaller: V. A. Zalgaller. Extremal problems on the convex hull of a space curve. Algebra i Analiz, 8(3):1–13, 1996....
• 7,057
Accepted

### Intuition behind the Dehn Invariant

The Dehn functional of a polyhedron $P$ with edge lengths $\ell_i$ and exterior dihedral angles $\theta_i$ is $D(P) = \sum_i \ell_i \otimes_{\mathbb Q} (\theta_i\, \mathrm{mod}\, \mathbb{Q}\pi)$. It ...
• 6,277

### Structures of the space of neural networks

I would like to argue that the space of neural networks is a category with finite products, or more concretely a Lawvere theory. This expresses an important piece of structure, namely how neural ...
• 5,967
Accepted

### Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry

This paper is contained in the 1-st volume of Aleksandrov's Selected works. This has an English translation: Alexandrov, A. D. Reshetnyak, Yu. G. (ed.) Selected works. Part 1: Selected scientific ...
• 89.7k
Accepted

### Does every Banach space admit a continuous (not necessarily equivalent) strictly convex norm?

If $\|\cdot\|_1$ is a continuous strictly convex norm on $(X,\|\cdot\|_0)$, then $\|x\|_2=\|x\|_0 + \|x\|_1$ defines an strictly convex norm on $X$ equivalent to $\|\cdot\|_0$. Therefore, a space like ...
• 4,301
Accepted

• 4,763

### Conditions for including cones

Iosif Pinelis already gave a nice solution to show that the answer is "no" for sets of infinitely many vectors, and thus for $N$ very large. I'll show that the answer is also "no" ...
• 5,765