In the paper *Surface sampling and the intrinsic Voronoi diagram (2008)*, Ramsay Dyer defines the *Gaussian curvature radius* at a point $x$ of a surface $S$ to be $\rho_K(x) = 1/\sqrt{K(x)}$ where $K(x)=\kappa_1(x) \kappa_2(x)$ is the Gaussian curvature at $x$.

Trying to track back the notion in Berger's *A panoramic view of Riemannian geometry*, and in Lee's *Riemannian manifolds* and in Chavel's *Riemannian Geometry* yielded nothing.

My question is two-folded:

- Where can I find more information about this notion?
- Is there a reason not to define it as $\rho_K(x) = 1/|K(x)|$? Otherwise, this definition is only valid for non-negatively curved surfaces.

**EDIT** As pointed out by Deane Yang, there is no sense in the definition I suggested. Nevertheless, if one wants to relate the Gaussian curvature to a radius (for either negatively or positively curved surfaces) how about this alternative: $\rho_{K}(x)=1/\sqrt{|K(x)|}$?