On the relation between the Jackson and Jung constants of the spaces $L_ p$

For any infinitely metrizable compact Abelian group $G$, $1\leqslant p\leqslant q<\infty$, $n\in\mathbb N$, the following relations are proved:
$$ K_{pq}(G,n,G)=d_{pq}(G,n,G)=J(L_p(G),L_q(G))=\varkappa_{pq}, $$
where $K_{pq}(G,n,G)$ is the largest Jackson constant in the approximation of the system of characters by polynomials of order $n$, $d_{pq}(G,n,G)$ is the best Jackson constant, $J(L_p(G),L_q(G))$ is the Jung constant of the pair of real spaces $(L_p(G),L_q(G))$, and
$$ \begin{aligned} \varkappa_{pq}^q&=\sup\biggl\{\inf_c\int_0^1|f(x)-c|^q\,dx \\ &\qquad\qquad\times\biggl|\int_0^1\int_0^1|f(x)-f(y)|\biggr|^p\,dx\,dy\le1,\ f\in L_q[-1,1]\biggr\}. \end{aligned} $$