  We establish computational results concerning the Lagrangian capacity from
"Cieliebak and Mohnke - Punctured holomorphic curves and Lagrangian
embeddings". More precisely, we show that the Lagrangian capacity of a
4-dimensional convex toric domain is equal to its diagonal. The proof involves
comparisons between the Lagrangian capacity, the McDuff-Siegel capacities from
"McDuff and Siegel - Symplectic capacities, unperturbed curves, and convex
toric domains", and the Gutt-Hutchings capacities from "Gutt and Hutchings -
Symplectic capacities from positive S1-equivariant symplectic homology".
Working under the assumption that there is a suitable virtual perturbation
scheme which defines the curve counts of linearized contact homology, we extend
the previous result to toric domains which are convex or concave and of any
dimension. For this, we use the higher symplectic capacities from "Siegel -
Higher symplectic capacities". The key step is showing that moduli spaces of
asymptotically cylindrical holomorphic curves in ellipsoids are transversely
cut out.
