Abstract : We consider a model of a $D$-dimensional tethered manifold interacting by excluded volume in $\mathbb{R}^d$ with a single point. By use of intrinsic distance geometry, we first provide a rigorous definition of the analytic continuation of its perturbative expansion for arbitrary $D$, 0 < $D$ < 2. We then construct explicitly a renormalization operation R, ensuring renormalizability to all orders. This is the first example of mathematical construction and renormalization for an interacting extended object with continuous internal dimension, encompassing field theory.
