The authors consider the semilinear polyharmonic Dirichlet problem \[ \begin{aligned} (-\Delta)^m u= f(u)\quad &\text{in }B,\\ u= {\partial u\over\partial r}=\cdots= {\partial^{m-1} u\over\partial r^{m-1}}= 0\quad &\text{on }\partial B.\end{aligned} \] Here \(B\) is the unit ball in \(\mathbb{R}^n\), \(r= |x|\) is the radial variable and \(f: [0,\infty)\to[0,\infty)\) is continuous and non-decreasing. Their main result is that every strong positive solution \(u\in H^m_0(B)\cap L^\infty(B)\) is radially symmetric and strictly decreasing in the radial variable. To prove this the authors adapt the method of moving planes using estimates for the Green function of the polyharmonic operator. They also consider minimizers for the compact embeddings \(H^m_0(B)\cap L^\infty(B)\subset L^p(B)\), that is functions \(u\) for which the best embedding constants are achieved. The minimizers are radially symmetric and monotone in the radial variable. For embeddings into weighted spaces with a radially symmetric weight function, they show that minimizers are axially symmetric. This result is sharp since examples are given of minimizers without radial symmetry.