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For any nonempty, compact and fiberwise convex set $K$ in $T^*\mathbb{R}^n$,we prove an isomorphism between symplectic homology of $K$ and a certainrelative homology of loop spaces of $\mathbb{R}^n$. We also prove a formulawhich computes symplectic homology capacity (which is a symplectic capacitydefined from symplectic homology) of $K$ using homology of loop spaces. Asapplications, we prove (i) symplectic homology capacity of any convex body isequal to its Ekeland-Hofer-Zehnder capacity, (ii) a certain subadditivityproperty of the Hofer-Zehnder capacity, which is a generalization of a resultpreviously proved by Haim-Kislev.

Symplectic homology of fiberwise convex sets and homology of loop spaces