Regularity in shape optimization under convexity constraint

This paper is concerned with the regularity of shape optimizers a class isoperimetric problems under convexity constraint. We prove that minimizers sum perimeter and perturbative term, among convex shapes, are $$C^{1,1}$$ -regular. To end, we define notion quasi-minimizer fitted to context show any such The proof relies on cutting procedure which was introduced similar results in calculus variations context. Using penalization method able treat volume constraint, showing same this case. go through some examples taken from PDE theory, when term type, large fit into our -regularity result. Finally provide counter-example cannot expect higher general.