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We establish computational results concerning the Lagrangian capacity,originally defined by Cieliebak-Mohnke. More precisely, we show that theLagrangian capacity of a 4-dimensional convex toric domain is equal to itsdiagonal. Working under the assumption that there is a suitable virtualperturbation scheme which defines the curve counts of linearized contacthomology, we extend the previous result to any convex or concave toric domain.This result gives a positive answer to a conjecture of Cieliebak-Mohnke for theLagrangian capacity of the ellipsoid.

On the Lagrangian capacity of convex or concave toric domains