DOI: 10.68381/jca28042 ISSN: 0944-6532
On the Linear Structures Induced by the Four Order Isomorphisms Acting on Cvx
0
(ℝ
n
)
Dan I. Florentin, Alexander Segal
It is known that the volume functional
\,\phi\mapsto\int e^{-\phi}\,
ϕ
↦
∫
e
−
ϕ
satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on
{\rm{Cvx}}_0(\mathbb{R}^n)
C
v
x
0
(
R
n
)
. In this note we define the fourth linear structure on
{\rm{Cvx}}_0(\mathbb{R}^n)
C
v
x
0
(
R
n
)
as the pullback of the standard linear structure under the
\cal J
J
transform. We show that, interpolating with respect to this linear structure, no concavity or convexity inequalities hold, and prove that a quasi-convexity inequality is violated only by up to a factor of
2
2
. We also establish all the order relations which the four different interpolations satisfy.