Ahlfors regularity of continua that minimize maxitive set functions

The primary objective of this paper is to establish the Ahlfors regularity of minimizers of set functions that satisfy a suitable maxitive condition on disjoint unions of sets. Our analysis focuses on minimizers within continua of the plane with finite 1-dimensional Hausdorff measure. Through quantitative estimates, we prove that the length of a minimizer inside the ball centered at one of its points is comparable to the radius of the ball. By operating within an abstract framework, we are able to encompass a diverse range of entities, including the inradius of a set, the maximum of the torsion function, and spectral functionals defined in terms of the eigenvalues of elliptic operators. These entities are of interest for several applications, such as structural engineering, urban planning, and quantum mechanics.