# Tagged Questions

**2**

votes

**1**answer

64 views

### Characterization of the medial axis of a surface

I would like to know if the following "characterization" of the medial axis of a surface is correct, and if so, how to prove it.
Let $S$ be a continuous, piecewise smooth, compact surface embedded in ...

**6**

votes

**0**answers

165 views

### Linked circles in R3

Two circles in 3-D are linked iff each one passes through the interior of the other.
There are $N$ points in 3-D in general position (no four lie on a plane). Each triple of points defines a unique ...

**2**

votes

**0**answers

74 views

### Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear ...

**8**

votes

**1**answer

216 views

### Fold-and-cut problem in three dimensions

The fold-and-cut theory states that "Any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include ...

**11**

votes

**3**answers

601 views

### What fraction of n-point sets in the unit ball have diameter smaller than 1?

This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...

**11**

votes

**0**answers

392 views

### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...

**9**

votes

**2**answers

527 views

### Discrete Morse function from smooth one

Suppose I have a smooth manifold $M$ and a smooth Morse function $f$ on $M$. Is there a standard way to replace $M$ with a cell complex and $f$ with a discrete Morse function such the resulting ...

**21**

votes

**1**answer

987 views

### Building a genus-$n$ torus from cubes

I wonder if this has been studied:
What is the fewest number of unit cubes
from which one can build an $n$-toroid?
The cubes must be glued face-to-face,
and the boundary of the resulting ...

**1**

vote

**2**answers

294 views

### Can a set of tetrahedra glued together by a common vertex be isometrically embedded in R^4?

A collection of triangles with a common vertex $A_1VA_2$, $A_2VA_3$, ... $A_NVA_1$ with specified side lengths can be isometrically embedded in $R^2$ provided the angles around $V$ add up to $2\pi$. ...

**2**

votes

**0**answers

184 views

### Is there any symmetric unshellable triangulation of a tetrahedron?

M. Rudin gave a triangulation of a tetrahedron with the property that after any small tetrahedron removed, the remaining part is not homeomorphic to a ball (An unshellable triangulation of a ...