On covering a ball by congruent subsets in normed spaces

We consider the covering of a ball in certain normed spaces by its congruent subsets and show that if the finite number of sets is not greater than the dimensionality of the space, then the centre of the ball either belongs to the interior of each set, or doesn’t belong to the interior of any set. We also provide some examples when it belongs to the interior of exactly one set. These are the specific cases of the modified problem originally posed for dissection.