Large-volume open sets in normed spaces without integral distances
  We study open sets $\mathcal{P}$ in normed spaces $X$ attaining a large volume while avoiding pairs of points at integral distance. The proposed task is to find sharp inequalities for the maximum possible $d$-dimensional volume. This problem can be viewed as an opposite to known problems on point sets with pairwise integral or rational distances. 
Sascha Kurz and Valery Mishkin
math.MG