We consider the problem of identifying the worst point-symmetric shape for
covering n-dimensional Euclidean space with lattice translates. Here we focus
on the dimensions where the thinnest lattice covering with balls is known and
ask whether the ball is a pessimum for covering in these dimensions compared to
all point-symmetric convex shapes. We find that the ball is a local pessimum in
3 dimensions, but not so for 4 and 5 dimensions