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Looks like Saúl RM has already achieved an arbitrarily high distance. Here's a simpler construction that at least gets distance $> 1$. The square has vertices $(\pm1/2, \pm1/2)$. The vertices near th …

There are plenty of $(C,P)$ such that all sections through $P$ have the same area. The construction readily adapts to higher moments about $C$ but not to perimeter. The existence of a single convex …

David Speyer writes "the only way this strategy could work is if we improve the $2$ in the previous paragraph to $\pi$." That improvement follows. Suppose $f < 0$ and $f + f'' \leq 0$ on $(0,c)$, with …

I doubt that there is a simple characterization. In any case conditions (1,2,3) are not sufficient. For example, if $n=2$ then $S$ cannot be ${\bf Z}_{\geq 100}$, and there are similar counterexamples …

Counterexample: for $(m,n)=(3,4)$ we can get a regular octahedron as the intersection of the tesseract $\Delta_4$ with the hyperplane $x_1+x_2 = x_3+x_4$. [It's easier to think of the equivalent but …

For $n=3$ this question was asked in 1996 by James Propp, conjecturing that the answer is Yes. (I got the reference from this page, which concerns the very special case of a rectangular box in ${\bf …

It seems that for large $n$ there will be an upper bound $c_n \alpha$ with $c_n$ asymptotic to $\frac{2}{9n}$ (and thus a bit better than the $\frac{1}{6n}$ suggested by the value of $c_1$). The asym …

Hermite normal form (HNF) should work much as you suggest. Translate $v_0$ to 0. Let $L = \bigoplus_{i=1}^n {\bf Z} v_i$. HNF gives an explicit decomposition of $G := {\bf Z}^n / L$ as a direct sum …