  Firstly, this paper establishes useful forms of the remainder term of
Hardy-type inequalities on general domains where the weights are functions of
the distance to the boundary. For weakly mean convex domains we use the
resulting identities to establish nonexistence of extremizers for and improve
known sharp Hardy inequalities. Secondly, we establish geometrically
interesting remainders for the Davies-Hardy-Tidblom inequalities for the mean
distance function, as well as generalize and improve several Hardy type
inequalities in the spirit of Brezis and Marcus and spectral estimates of
Davies. Lastly, we apply our results to obtain Sobolev inequalities for
non-regular Riemannian metrics on geometric exterior domains.
