Among the various width-type constraints considered in shape optimization, the most classical is the maximal width (or diameter), which describes how "large" an admissible set can be. Less commonly studied - but equally significant - is the minimal width, which instead measures the "thickness" of admissible shapes.In this seminar, I will present some recent results on the optimisation, also in quantitative form, of geometric and spectral-type functionals under diameter or thickness constraint. The talk is based on joint works with Ilias Ftouhi (Nimes, France), Antoine Henrot (Nancy, France), Giorgio Saracco (Ferrara, Italy), and Davide Zucco (Turin, Italy).
