# Tag Info

1 vote

• 43.6k

### Does $\mathbb Z^n$ contain $A_n$?

The even sublattice of $\mathbb Z^n$ is the root lattice $D_n$ which does not contain $A_n$ for $n\geq 4$. ($D_2=A_1\oplus A_1$ and $D_3=A_3$ are generally omitted in lists of root-systems) This ...
• 17.3k
Accepted

• 6,881
Accepted

### A triangle comparison in CAT(0) spaces

No, it is not true. Suppose $X$ glued from two solid plane triangles $\blacktriangle p x \bar y$ and $\blacktriangle y x\bar y$, the angle at $y$ is obtuce and the total angle at $\bar y$ is reflex. ...
• 43.6k
Accepted

### Forming paper bags that can 'trap' 3D regions of max surface area

Yes, if $S$ is convex, then you can form what is called in Geometric Folding Algorithms (p. 382) a perimeter-halving as follows. Pick any point $x$ on $\partial S$, and another point $y \in \partial S$...
• 149k
Accepted

### Characterization of convexity by connectedness of hyperplane sections

Suppose $n\geqslant 2$. Let $S$ be the boundary of $\tfrac12$-neighborhood of the unit $n$-sphere in $\mathbb{R}^{2{\cdot}n+1}$. (Note that $S$ is homeomorphic to $\mathbb{S}^n\times\mathbb{S}^n$.) ...
• 43.6k
There has been recent progress concerning finitary integer distance sets in the plane. Greenfeld, Iliopoulou, and Peluse prove that if $P\subset [-N,N]^2$ is an integer distance set not contained ...