  Optimal weighted Sobolev-Lorentz embeddings with homogeneous weights in open
convex cones are established, with the exact value of the optimal constant.
These embeddings are non-compact, and this paper investigates the structure of
their non-compactness quantitatively. Opposite to the previous results in this
direction, the non-compactness in this case does not occur uniformly over all
sub-domains of the underlying domain, and the problem is not translation
invariant, and so these properties cannot be exploited here. Nevertheless, by
developing a new approach based on a delicate interplay between the size of
suitable extremal functions and the size of their supports, the exact values of
the (ball) measure of non-compactness and of all injective strict s-numbers (in
particular, of the Bernstein numbers) are obtained. Moreover, it is also shown
that the embedding is not strictly singular.
