Kaba wrote: > Kaba wrote: > > Therefore, sharpening the problem, the problem is to characterize when > > triangle inequality holds if B is indefinite. > > Let's not forget: > > Triangle inequality holds for f > <=> > The sublevel sets of f are convex (f is quasiconvex). > <=> > The f is convex. > > Here quasi-convexity happens to be equivalent to convexity although I am > not sure why (possibly because of homogenuity of f).
Scrap that. Quasiconvexity does not imply triangle inequality or convexity here. The following hold:
Triangle inequality holds for f <=> The f is convex
The f is convex ==> The sublevel sets of f are convex (f is quasiconvex).