Abstract. 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.

We introduce the concept of a visual boundary of a domain Ω ⊂ R n and show that the weighted Hardy inequality Ω |u| p dΩ β−p C Ω |∇u| p dΩ β , where dΩ(x) = dist(x, ∂Ω), holds for all u ∈ C ∞ 0 (Ω) with exponents β < β0 when the visual boundary of Ω is sufficiently large. Here β 0 = β0(p, n, Ω) is explicit, essentially sharp, and may even be greater than p − 1, which is the known bound for smooth domains. For instance, in the case of the usual von Koch snowflake domain the sharp bound is shown to be β 0 = p − 2 + λ, with λ = log 4/ log 3. These results are based on new pointwise Hardy inequalities.

We obtain estimates for the nonlinear variational capacity of annuli in weighted R n and in metric spaces. We introduce four different (pointwise) exponent sets, show that they all play fundamental roles for capacity estimates, and also demonstrate that whether an end point of an exponent set is attained or not is important. As a consequence of our estimates we obtain, for instance, criteria for points to have zero (resp. positive) capacity. Our discussion holds in rather general metric spaces, including Carnot groups and many manifolds, but it is just as relevant on weighted R n . Indeed, to illustrate the sharpness of our estimates, we give several examples of radially weighted R n , which are based on quasiconformality of radial stretchings in R n .

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