41
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 $\gamma$ ...

41
votes

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): ...

28
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 \...

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 ...

25
votes

Accepted

### 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 ...

24
votes

Accepted

### Acute triangles in "obtuse" polygons?

Take a very obtuse isosceles triangle and chop its acute angles.

23
votes

Accepted

### 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 ...

23
votes

Accepted

### 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 ...

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 ...

21
votes

Accepted

### 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 $\sum_{i=1}^4 a_i=1$ ...

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 ...

17
votes

Accepted

### 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 ...

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 $C$ ...

15
votes

Accepted

### 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 ...

14
votes

Accepted

### 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-...

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 (...

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 ...

14
votes

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 ...

14
votes

Accepted

### 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 ...

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....

13
votes

Accepted

### 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 ...

13
votes

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 ...

13
votes

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 ...

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 ...

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 ...

12
votes

Accepted

### 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^{...

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 $...

11
votes

### Approximating a convex disk by an ellipse

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 ...

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