Decomposition of balls in ${\mathbb{R}}^d$

We investigate the decomposition problem of balls into finitely many congruent pieces in dimension $d=2k$. In addition, we prove that the $d$ dimensional unit ball $B_d$ can be divided into finitely many congruent pieces if $d=4$ or $d\ge 6$. We show that the minimal number of required pieces is less than $20d$ if $d\ge 10$.