2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59349We deal with a family of functionals depending on curvatures and we prove for them compactness and semicontinuity properties in the class of closed and bounded sets which satisfy a uniform exterior and interior sphere condition. We apply the results to state an existence theorem for the Nitzberg and Mumford problem under this additional constraint.20 pages. To appear on Rendiconti del Seminario Matematico dell'Universita' di PadovaFunctional Analysistext