It is proven that the Löwner-Heinz inequality
‖
A
t
B
t
‖
≤
‖
A
B
‖
t
{\|A^{t}B^{t}\|\le \|AB\|^{t}}
, valid for all positive invertible operators
A
,
B
{A, B}
on the Hilbert space
H
{\mathcal H }
and
t
∈
[
0
,
1
]
{t\in [0,1]}
, has equivalent forms related to the Finsler structure of the space of positive invertible elements of
L
(
H
)
{\mathcal L (\mathcal H )}
or, more generally, of a unital
C
∗
{C^{*}}
-algebra. In particular, the Löwner-Heinz inequality is equivalent to some type of “nonpositive curvature" property of that space.