On symmetry of the extremal in some embedding theorems

We study the symmetry/asymmetry of functions providing sharp constants in the embedding theorems $\mathring W_2^r(-1,1)\hookrightarrow\mathring W_\infty^k(-1,1)$ for various $r$ and $k$. The sharp constants for all $r>k$ in the cases $k=4$ and $k=6$ are calculated explicitly as well.