In this paper we consider a family of convex sets in
R
n
\mathbf {R}^{n}
,
F
=
{
S
(
x
,
t
)
}
\mathcal {F}= \{S(x,t)\}
,
x
∈
R
n
x\in \mathbf {R}^{n}
,
t
>
0
t>0
, satisfying certain axioms of affine invariance, and a Borel measure
μ
\mu
satisfying a doubling condition with respect to the family
F
.
\mathcal {F}.
The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of
F
.
\mathcal {F}.
This is achieved by showing first a Besicovitch-type covering lemma for the family
F
\mathcal {F}
and then using the doubling property of the measure
μ
.
\mu .
The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to
F
.
\mathcal {F}.