3
votes
0answers
141 views

Packing space by cones: Translates best?

Let $C$ be a right circular cone, the convex hull of a unit-radius disk and a point directly above the disk center at height $h$.                 Is the ...
6
votes
1answer
152 views

Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem), "Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...
14
votes
0answers
420 views

An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
2
votes
1answer
210 views

Passing C through a slot

Question: Given a closed curve C, what will be the (bounds on) dimension of the interval it will pass through? i.e. which are the necessary and sufficient conditions for a planar compact set C to ...
2
votes
1answer
143 views

Equiprojective polyhedra

Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open, and which some might find intriguing. Define an ...
10
votes
1answer
455 views

Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$. Say that $S$ is an integer-distance set if every pair of points in $S$ is separated by an integer Euclidean distance. ...
9
votes
1answer
299 views

Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of volume vol$(K) = V$ inside the unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$. You are permitted to probe with a (one-dimensional) ...
22
votes
4answers
1k views

Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points ...
16
votes
2answers
969 views

Center of mass from the abstract point of view, or could the ancient Greeks invent modern analysis?

This is a very open-ended question, which may or may not have a perfect answer, and for which I have a few ideas but nothing like a clear picture. However, I guess it won't hurt to ask to see if ...
9
votes
1answer
480 views

Minimizing the excursion of a sum of unit vectors

I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose sum is zero: $$ v_1 + v_2 + \cdots + v_n = 0 \; .$$ Now I form the closed polygon $P$ in space by placing them head to tail. So the ...
9
votes
0answers
737 views

3-piece dissection of square to equilateral triangle?

At a workshop it was suggested that it likely remains an open problem whether or not there is a 3- or 2 -piece dissection of a square to an equilateral triangle. Can anyone confirm that this is ...
3
votes
0answers
356 views

Compact surfaces smoothly immerse in: $\mathbb{R}^4$ or $\mathbb{R}^5$?

I wonder if someone can clarify whether it is known that every closed, orientable surface (2-manifold) has a smooth isometric immersion in $\mathbb{R}^4$? This topic has been discussed rather ...
13
votes
2answers
536 views

What is known about Ulam's problem of metric spaces with isometric squares?

Background In the book Problems in Modern Mathematics, S. Ulam asks the following question: Suppose $A$ and $B$ are metric spaces, such that $A^2$ and $B^2$ equipped with the 2-metric $d((x_1, ...
29
votes
0answers
1k views

Shortest closed curve to inspect a sphere

Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and exterior to $S$ which has the property that every point $x$ on $S$ is visible to some point $y$ of ...
9
votes
6answers
1k views

Smallest area shape that covers all unit length curve

On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve. What are the bounds of the shape's area if this ...
24
votes
4answers
2k views

Is Lebesgue's “universal covering” problem still open?

The following problem has been attributed to Lebesgue. Let "set" denote any subset of the Euclidean plane. What is the greatest lower bound of the diameter of any set which contains a subset congruent ...
8
votes
1answer
1k views

Smallest dilation of a quadrilateral?

What is the smallest dilation of a quadrilateral in $\mathbb{R}^d$? This may be an open problem, which I know is verboten on MO. So my question is: Is this indeed open? It will take me some time to ...
55
votes
0answers
3k views

Volumes of Sets of Constant Width in High Dimensions

Background The n dimensional Euclidean ball of radius 1/2 has width 1 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them ...
17
votes
1answer
1k views

Coiling Rope in a Box

What is the longest rope length L of radius r that can fit into a box? The rope is a smooth curve with a tubular neighborhood of radius r, such that the rope does not self-penetrate. For an open ...
27
votes
3answers
2k views

Is the ratio Perimeter/Area for a finite union of unit squares at most 4?

Update: As I have just learned, this is called Keleti's perimeter area conjecture. Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
12
votes
3answers
983 views

distance regular metric spaces

A metric space (V,d) will be called distance regular if for every distances a>0, b, c a nonnegative integer p(a,b,c) is defined, so that whenever d(B,C)=a, there are precisely p(a,b,c) points A ...
10
votes
4answers
870 views

Diameter of universal cover

Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric). What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or ...
25
votes
6answers
2k views

When shorter means smaller?

Assume a convex figure $F\subset \mathbb R^2$ satisfies the following property: if $f:F\to \mathbb R^2$ is a distance-non-increasing map then its image $f(F)$ is congruent to a subset of $F$. Is it ...
7
votes
2answers
573 views

What is known about polyhedra nets that allow overlapping?

It is an open problem that the net of any convex polyhedron can be unfolded onto a flat plane with no overlapping. Is anything known if we allow x faces to overlap? For example, is it known if any ...
39
votes
12answers
2k views

Can a discrete set of the plane of uniform density intersect all large triangles?

Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximatively equal to the area of the disc. Does the complement of S necessarily contain ...