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44 votes
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 $\gamma$ ...
Mohammad Ghomi's user avatar
41 votes
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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): ...
Joseph O'Rourke's user avatar
29 votes
Accepted

Reference to a conjecture on unit vectors in Euclidean space

That isn't a conjecture but a routine exercise assigned after the students learn about Bang's solution of the Tarski plank problem. The proof goes in 2 steps: 1) Consider all sums $\sum_j \...
fedja's user avatar
  • 60.5k
27 votes
Accepted

Concurrent normals conjecture

Let me address the specific complaint of that review. The situation is the following. Our (bounded, open) convex set is denoted $K\subseteq\mathbb R^n$ with closure $\overline K$, and we consider ...
John Pardon's user avatar
  • 18.4k
25 votes
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Abstract definition of convex set

There has been a bunch of work along these lines, and I think the idea has been rediscovered several times. I suggest looking at the papers of Anna Romanowska, who refers to them as "barycentric ...
arsmath's user avatar
  • 6,750
24 votes
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Acute triangles in "obtuse" polygons?

Take a very obtuse isosceles triangle and chop its acute angles.
Anton Petrunin's user avatar
23 votes
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Shortest path connecting two opposite points on a cube

Consider the sphere with equator 4. Divide it into spherical cubes, the central projections from an inscribed cube. Note that the exponential map from tangent plane to the sphere is short. Note also ...
Anton Petrunin's user avatar
23 votes
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Average measure of intersection of a convex region with its translate

The following proposition answers OP's question regarding the upper bound of $$\tau(C) \Doteq f(C)/\lambda^2(C).$$ Let $B_n$ be the closed Euclidean unit ball of $\mathbb{R^n}$ centred at $0$, that is ...
Luc Guyot's user avatar
  • 7,588
21 votes

Shortest closed curve to inspect a sphere

I have recently finished a paper called The length, width, and inradius of space curves where it is shown that the length $L$ of any closed curve $\gamma\colon[a,b]\to \mathbf{R}^3$ inspecting the ...
Mohammad Ghomi's user avatar
21 votes
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Can we always shift two disjoint convex bodies a little bit to decrease the volume of their convex hull?

This fails already for $d=3$. Consider a tetrahedron, e.g. the convex hull of the points $v_1,v_2,v_3,v_4$. Let $K$ be the closed subset consisting of $\sum_{i=1}^4 a_i v_i$ with $\sum_{i=1}^4 a_i=1$ ...
Will Sawin's user avatar
  • 139k
17 votes

How to make a sandwich from just one piece of bread?

This won't be a complete answer to Q2, but something of a starting point, at least for all two-dimensional convex $P$. Assuming for simplicity that the area of $P$ is 1, $P$ contains a parallelogram ...
Wlodek Kuperberg's user avatar
17 votes
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Does Helly's theorem hold in the hyperbolic plane?

I don't understand your reference to the model (since the geometry of the hyperbolic plane does not depend on any model), but, in fact, the Beltrami-Klein model demonstrates that any qualitative ...
Igor Rivin's user avatar
  • 95.7k
15 votes

Shortest path connecting two opposite points on a cube

[This is an attempt to explain the details in Anton Petrunin's answer to this question, since the comments suggest that a number of people have found it hard to understand as it was written.] Let $C$ ...
Gro-Tsen's user avatar
  • 30.6k
15 votes
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Two questions on the permutohedron

The number of integer points in $P_n$ is the number of forests on $[n]$; see Section 3 of Stanley's Decompositions of rational convex polytopes. In fact you can see there a simple description of its ...
Sam Hopkins's user avatar
  • 23.1k
15 votes
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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 ...
fedja's user avatar
  • 60.5k
14 votes
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covering convex sets by round balls

Yes. Any point $y$ in the convex hull of $x$'s is a barycenter of some non-negative masses $m_i$ in $x_i$, $\sum m_i=1$, $y=\sum m_i x_i$. Point $y$ minimizes the moment of inertia $I(p)=\sum m_i |p-...
Fedor Petrov's user avatar
14 votes

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 (...
Ivan Izmestiev's user avatar
14 votes
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 ...
Alexandre Eremenko's user avatar
14 votes
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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 ...
Gerald Edgar's user avatar
  • 40.6k
14 votes
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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 ...
Mohammad Ghomi's user avatar
13 votes

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....
Mohammad Ghomi's user avatar
13 votes
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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 ...
Ivan Izmestiev's user avatar
13 votes

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 ...
Tobias Fritz's user avatar
  • 5,967
13 votes
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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 ...
Alexandre Eremenko's user avatar
13 votes
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 ...
M.González's user avatar
  • 4,301
12 votes
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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 $...
Gerald Edgar's user avatar
  • 40.6k
12 votes
Accepted

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 ...
Ivan Izmestiev's user avatar
12 votes
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Is Minkowski sum of boundary convex again?

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^{...
Guillaume Aubrun's user avatar
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" ...
Nathaniel Johnston's user avatar
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. ...
Suvrit's user avatar
  • 28.4k

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