Beyond traditional curvature-dimension I: New model spaces for isoperimetric and concentration inequalities in negative dimension

We study the isoperimetric, functional and concentration properties of n-dimensional weighted Riemannian manifolds satisfying the Curvature- Dimension condition, when the generalized dimension N is negative and, more generally, is in the range N ∈ (−∞, 1), extending the scope from the traditional range N ∈ [n,∞]. In particular, we identify the correct one-dimensional model-spaces under an additional diameter upper bound and discover a new case yielding a single model space (besides the previously known N-sphere and Gaussian measure when N ∈ [n,∞]): a (positively curved) sphere of (possibly negative) dimension N ∈ (−∞, 1). When curvature is non-negative, we show that arbitrarily weak concentration implies an N-dimensional Cheeger isoperimetric inequality and derive various weak Sobolev and Nash-type inequalities on such spaces. When curvature is strictly positive, we observe that such spaces satisfy a Poincaré inequality uniformly for all N ∈ (−∞, 1 − ε]. and enjoy a two-level concentration of the type exp(−min(t, t²)). Our main technical tool is a generalized version of the Heintze–Karcher theorem, which we extend to the range N ∈ (−∞, 1).