Given convex
u\in C(\bar{\Omega})
u
∈
C
(
Ω
ˉ
)
with Monge-Ampère measure
Mu
M
u
, and finite Borel measures
\mu
μ
and
\nu
ν
satisfying
\mu + \nu = Mu
μ
+
ν
=
M
u
, consider the problem of determining a ‘splitting’
u=v+w
u
=
v
+
w
for
u
u
where
v,w \in C(\bar{\Omega})
v
,
w
∈
C
(
Ω
ˉ
)
are convex functions satisfying
Mv=\mu
M
v
=
μ
,
Mw=\nu
M
w
=
ν
, so that
Mu=M(v+w)=Mv + Mw
M
u
=
M
(
v
+
w
)
=
M
v
+
M
w
. It is shown that although this problem is not in general solvable, a best
L^p
L
p
approximation
v^\ast+w^\ast
v
∗
+
w
∗
for
u
u
may always be found. In particular, letting
U={\rm sup}_{(v,w)\in {\cal F}}~(v+w)
U
=
s
u
p
(
v
,
w
)
∈
F
(
v
+
w
)
, there exist optimal sums
v^\ast+w^\ast
v
∗
+
w
∗
achieving
{\rm inf}_{(v,w)\in {\cal F}}~\|u-(v+w)\|_p
i
n
f
(
v
,
w
)
∈
F
∥
u
−
(
v
+
w
)
∥
p
and
{\rm inf}_{(v,w)\in {\cal F}}~\|U-(v+w)\|_p
i
n
f
(
v
,
w
)
∈
F
∥
U
−
(
v
+
w
)
∥
p
,
p\ge 1
p
≥
1
, for appropriately constrained classes
{\cal F}
F
of feasible pairs
(v,w)
(
v
,
w
)
of convex functions satisfying
Mv=\mu
M
v
=
μ
,
Mw=\nu
M
w
=
ν
and
v+w=u
v
+
w
=
u
on
\partial\Omega
∂
Ω
. Moreover,
U
U
may be written as
U=\bar{v}+\bar{w}
U
=
v
ˉ
+
w
ˉ
within
\bar{\Omega}
Ω
ˉ
,
(\bar{v},\bar{w})\in {\cal F}
(
v
ˉ
,
w
ˉ
)
∈
F
. The analysis depends upon basic properties of convex functions and the measures they determine. We also consider the related problem of characterizing functions
u\in W^{2,n}(\Omega)
u
∈
W
2
,
n
(
Ω
)
which may be realized as differences
u=v-w
u
=
v
−
w
of convex functions
v,w\in W^{2,n}(\Omega)
v
,
w
∈
W
2
,
n
(
Ω
)
with
Mu=Mv-Mw
M
u
=
M
v
−
M
w
. Here
Mu
M
u
is the signed measure defined by
dMu={\rm det}~D^2u\,dx
d
M
u
=
d
e
t
D
2
u
d
x
. Letting
U^-={\rm sup}_{(v,w)\in {\cal F}}(v-w)
U
−
=
s
u
p
(
v
,
w
)
∈
F
(
v
−
w
)
and
U_-={\rm inf}_{(v,w)\in {\cal F}}(v-w)
U
−
=
i
n
f
(
v
,
w
)
∈
F
(
v
−
w
)
, we show that optimal differences
v^\ast-w^\ast
v
∗
−
w
∗
exist for the problems
{\rm inf}_{(v,w)\in {\cal F}}~\|u-(v-w)\|_p
i
n
f
(
v
,
w
)
∈
F
∥
u
−
(
v
−
w
)
∥
p
,
{\rm inf}_{(v,w)\in {\cal F}}~\|U^--(v-w)\|_p
i
n
f
(
v
,
w
)
∈
F
∥
U
−
−
(
v
−
w
)
∥
p
and
{\rm inf}_{(v,w)\in {\cal F}}~\|U_- -(v-w)\|_p
i
n
f
(
v
,
w
)
∈
F
∥
U
−
−
(
v
−
w
)
∥
p
. Also,
U^-=v^--w^-
U
−
=
v
−
−
w
−
and
U_-=v_--w_-
U
−
=
v
−
−
w
−
for appropriate pairs
(v^-,w^-),(v_-,w_-)\in {\cal F}
(
v
−
,
w
−
)
,
(
v
−
,
w
−
)
∈
F
. Finally, the relaxed problem of finding
v+w=u
v
+
w
=
u
for general
Mv
M
v
and
Mw
M
w
with
Mv+Mw = Mu
M
v
+
M
w
=
M
u
(no fixed
\mu
μ
and
\nu
ν
), is considered. Topological properties of the collection of these relaxed splitting pairs
(v,w)
(
v
,
w
)
, and those for the unrelaxed problem, for a given
u
u
, are developed.