22
votes

Accepted

### Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

A counterexample is given by the following five points:
$$(0,0),(1,0),
\Big(-\frac{64867}{77629},\frac{3389}{60094}\Big),
\Big(\frac{5981}{56176},\frac{32211}{34172}\Big),
\Big(\frac{5925}{117812},-\...

11
votes

### Smallest sphere containing three tetrahedra?

Unclear this is best, but: $r=\sqrt{3}/2$.
JukkaKohonen's suggestion:
I made no attempt to optimize, but this certainly shows
the smallest sphere has radius strictly less than $\sqrt{3}/2$.

10
votes

### Has there been any progress on Conway's and Soifer's shortest paper?

Jineon Baek and Seewoo Lee recently posted to the arXiv a paper claiming to prove the conjecture in the case that every small triangle in the cover has edges parallel to the large triangle. (The ...

3
votes

### Is there a regular pentagon with a rational point on each edge?

This is an attempt giving some partial results.
Set $s=\sin(2\pi/5)$ and $c=2s^2 - 3/2=\cos(2\pi/5)$. We show that if there is a positive answer, then the slopes of the edges are contained in the ...

3
votes

Accepted

### On the moment of inertia of planar convex regions and possible special nature of circular disks

The moment of inertia is a quadratic form: for a region $R$ with respect to a line through the origin in the direction of a unit vector $v$, it is given by $\int_{R}(v\cdot x)^2\,dx=(Av)\cdot v$ for a ...

2
votes

### Algorithm for grouping tetrahedra from Voronoi diagram

Chapter 27 of The Handbook of Discrete and Computational Geometry
enumerates the complexity of known Delaunay triangulation algorithms of any dimension (see Table 27.2.1 there).
For 3D Delaunay the ...

2
votes

### Tiling a rectangle with all simply connected polyominoes of fixed size

I believe $n = 17$ is also impossible for a similar reason as $n \geq 18$. According to a computer search, there are $219$ hole-less 17-ominoes that create a $4\times 3$ rectangular cavity, but only $...

2
votes

Accepted

### On 'special' points on uniform planar convex regions defined in terms of moment of inertia

$\newcommand{\la}{\lambda}\newcommand{\de}{\delta}\newcommand{\ep}{\delta}$The answer is as follows:
Yes, if the special point is allowed to be not in $C$, and then the moments of inertia of $C$ ...

1
vote

Accepted

### To find the convex planar region minimizing diameter when area and perimeter are given

The 2000 paper Inequalities for Convex Sets, by Paul R. Scott and Poh Way Awyong, lists various inequalities on 2D convex bodies as described here; for your question you can see the state of the ...

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