Here's one that I've found very vexing: Let A(d) denote the least surface area of a cell that tiles R^d via the translations in Z^d. Improve on the following bounds:It's a meta post !!! And by the way, there's no such thing as "JUST a problem in geometry". Harumph !!
sqrt(pi e / 2) sqrt(d) <= A(d)/2 <= d - exp(-Omega(d log d)).
(It's a little more natural to consider A(d)/2 than A(d).)
The lower bound is by considering a volume-1 sphere, the upper bound by monkeying around a little with a cube's corner.
Any proof of A(d)/2 >= 10sqrt(d) or A(d)/2 <= d/10 would be very interesting.
For motivation: Either pretend I posted on Suresh's blog (so it's just a problem in geometry); or -- it's related to finding the best rate for parallel repetition of a simple family of 2P1R games.