Dimensions, Whitney covers, and tubular neighborhoods
  Working in doubling metric spaces, we examine the connections between different dimensions, Whitney covers, and geometrical properties of tubular neighborhoods. In the Euclidean space, we relate these concepts to the behavior of the surface area of the boundaries of parallel sets. In particular, we give characterizations for the Minkowski and the spherical dimensions by means of the Whitney ball count. 
Antti K\"aenm\"aki, Juha Lehrb\"ack, Matti Vuorinen
math.MG