# 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 $\gamma$ ...
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): ...
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### 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 (...
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### 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 ...
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### 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 ...
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### Convex functions in convex sets

Zachary Chase said that he is still interested, so I'll make a post. It is a bit on the long side and I'll need to draw a few pictures to make it comprehensible, so today I'll just post a ...

### 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....
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### Tighter Caratheodory on the moment curve?

The answer is yes for all dimensions. An old theorem by Fenchel states that for a compact set $K$ in $\mathbb{R}^n$ every point in the convex hull can be either written as a convex combination of at ...
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### 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 ...
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### 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 ...
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### 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 ...
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### What is known about sufficient conditions for the rigidity of a convex surface?

Answer to question 1: proper compact subsets of convex surfaces are infinitesimally flexible. In his book Pogorelov proves infinitesimal flexibility of the graph of a convex function over a strictly ...
Yes, $\partial C + \partial C$ is convex since it equals $2C$. Equivalently, every point in $z \in C$ is a midpoint of two boundary points. This is obvious if $z \in \partial C$. Otherwise, let $f :S^{... 12 votes ### Conditions for including cones Iosif Pinelis already gave a nice solution to show that the answer is "no" for sets of infinitely many vectors, and thus for$N$very large. I'll show that the answer is also "no" ... 11 votes Accepted ### What is the "positive part" of the unit ball in$M_n(R)$? I'm a bit late in answering this. But in case there is still interest, please have a look at: Saunderson, Parrilo, Willsky. Semidefinite descriptions of the convex hull of rotation matrices, SIAM J. ... 11 votes Accepted ### weak*-closed convex = closed convex? No it is false in general. Yes, there is a class of spaces where it is true: these are exactly the reflexive spaces. Suppose$X$is not reflexive. Then considering$X$embedded into$X''$, we have$...
Not an answer, just an illustration to accompany the question. $K$ is an isosceles triangle with base $2$ and altitude $3$ (and so area $3$). First, I mistakenly computed the ellipse $E$ of any area ...