*preprint*

**Inserted:** 1 jun 2022

**Last Updated:** 1 jun 2022

**Year:** 2022

**Abstract:**

Recently, in the metric spaces, Le Donne and the author introduced the so-called intrinsically Lipschitz sections. The main aim of this note is to adapt Cheeger theory for the classical Lipschitz constants in our new context. More precisely, we define the intrinsic Cheeger energy from $L^2(Y)$ to $[1,+\infty],$ where $(Y,d_Y,\mm)$ is a metric measure space with $Y\subset \R$ and we characterize it in terms of a suitable notion of relaxed slope. In order to get this result, in more general context, we establish some properties of the intrinsically Lipschitz constants like the Leibniz formula, the product formula and the upper semicontinuity of the asymptotic in\-trin\-si\-cally Lipschitz constant. Finally, adapting the concept of relaxed slope in our intrinsic context, we define the so-called intrinsic relaxed slope in strong sense and we show that this new class is a vector space over $\R.$