We will discuss recent results establishing $L^p$-improving estimates for Radon-like operators which average functions over submanifolds of intermediate dimension (e.g., neither curves nor hypersurfaces). The methods are built around an $L^p$-adapted $TT^*T$ argument which is itself an instance of a Christ-type method of refinements. The resulting estimates are sharp up to loss of the endpoints and provide new insights into the vector field formulation of sharp curvature conditions