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 $\gamma$ ...
- 7,149
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): ...
- 151k
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 \...
- 61.9k
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 ...
- 18.7k
26
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 ...
- 6,860
24
votes
Accepted
Acute triangles in "obtuse" polygons?
Take a very obtuse isosceles triangle and chop its acute angles.
- 45k
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 ...
- 45k
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 ...
- 7,893
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 $\sum_{i=1}^4 a_i=1$ ...
- 148k
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 ...
- 7,276
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 ...
- 96.4k
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 $C$ ...
- 32.4k
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 ...
- 24.2k
15
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 ...
- 61.9k
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-...
- 109k
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 (...
- 6,337
14
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 ...
- 6,406
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 ...
- 91.8k
14
votes
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 ...
- 41.1k
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 ...
- 7,149
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 ...
- 6,337
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 ...
- 91.8k
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 ...
- 4,461
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 ...
- 6,337
12
votes
Accepted
Where to find some subset of Khovanskii's 15 proofs of the BKK theorem?
Khovanskii gives what he calls "the simplest proof" in section 4 of Newton Polyhedron, Hilbert Polynomial, and Sums of Finite Sets (1992).
More proofs are in
Newton polyhedra and the genus of ...
- 188k
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^{...
- 4,653
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" ...
- 6,351
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. ...
- 28.6k
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 ...
- 151k
11
votes
How to make a sandwich from just one piece of bread?
Restricting to the case where $P$ is triangular, if the angles of the triangle are $A$, $B$, $C$ and you fold the triangle along the bisector to the angle $C$, then assuming wlog that $A<B$, you ...
- 2,902
Only top scored, non community-wiki answers of a minimum length are eligible