In this talk, I will present some results obtained during my PhD, here at LAMA. I will talk about shape optimization problems under inradius constraint, focusing on the 2D case. I will discuss two approaches to tackle these problems, one analytical and the other numerical. First, I present a result obtained in collaboration with D. Bucur and G. Buttazzo. We show that the honeycomb structure is optimal for the Cheeger constant and the torsional rigidity. I will focus my presentation on the Cheeger case. Secondly, I explain a new numerical scheme implemented with B. Bogosel. We aim to numerically solve these problems with the added constraint of simple connectivity, in the framework of the levelset method.