We prove the following result: let
K\subseteq \mathbb{R}^N
K
⊆
R
N
be convex with nonempty interior,
X
X
a topological space and
f\colon K\times X\to\mathbb{R}
f
:
K
×
X
→
R
be concave and u.s.c. in the first variable and coercive and l.s.c. in the second. Then the (perturbed) strict minimax inequality
\sup_{\lambda\in K}\inf_{x\in X}f(\lambda,x)+g(\lambda)<\inf_{x\in X} \sup_{\lambda\in K}f(\lambda,x)+g(\lambda),
sup
λ
∈
K
inf
x
∈
X
f
(
λ
,
x
)
+
g
(
λ
)
<
inf
x
∈
X
sup
λ
∈
K
f
(
λ
,
x
)
+
g
(
λ
)
,
for some continuous concave
g\colon K\to\mathbb{R}
g
:
K
→
R
, is equivalent to the following condition on superdifferentials: if
F(\lambda)=\inf_X f(\lambda, x)
F
(
λ
)
=
inf
X
f
(
λ
,
x
)
, for some
\lambda\in\mathring{K}
λ
∈
K
˚
\partial F(\lambda)\setminus \bigcup_{\substack{x\in X\\ f(\lambda, x) =F(\lambda)}}\partial f(\lambda, x)\neq\emptyset.
∂
F
(
λ
)
∖
⋃
x
∈
X
f
(
λ
,
x
)
=
F
(
λ
)
∂
f
(
λ
,
x
)
≠
∅
.
As an application of this differential characterisation we prove a generalised version of a theorem of Ricceri, a criterion of regularity for marginal functions, and the fact that to check whether some perturbed minimax inequality holds, one can test with affine perturbation only.