Towards the strong Viterbo conjecture.

This paper is a step towards the strong Viterbo conjecture on the coincidence of all symplectic capacities on convex domains. Our main result is a proof of this conjecture in dimension 4 for the classes of convex and concave toric domains. The second result is that, in any dimension, $c_{1}^{\operatorname{Ekeland-Hofer}}(W)=c_1^{CH}(W)=c_{\operatorname{Viterbo}}(W)$ for all convex domains $W\subset\mathbb{R}^{2n}$. Moreover, if $W$ is a convex or concave toric domain $W=X_\Omega\subset\mathbb{R}^{2n}$, then $c_1^{CH}(X_\Omega)=c_{\operatorname{Gromov}}(X_\Omega)$.