  We consider a model of D-dimensional tethered manifold interacting by
excluded volume in 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, 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.
