A Bose type formula for the internal medial axis of an embedded manifold

We establish a three-dimensional variant of the Bose formula for the internal medial axis of a closed plane curve. We generalize the result to dimensions less than or equal to 6, and we apply the result to cut-loci, and spines of 3-manifolds. Also, we describe the difference between the main theorems and recent work of Sedykh.