We study isomeries on subsets of the positive cones of unital $C^*$algebras. We introduce a notion of a generalized gyrovector space. We show a geometric inequality :
$||| \log(a^{1/2}ba^{1/2})||| \le ||| \log a|||+ ||| \log b|||$

for every pair $a, b ∈ B(H)^{-1}_+$ and a complete uniform norm. As an application of the inequality we prove that certain subsets of positive cones are GGV. Applying a Mazur-Ulam theorem for GGV we get the forms of isometries.