Let the space
R
n
\mathbb R^n
be endowed with a Minkowski structure
M
M
(that is,
M
:
R
n
→
[
0
,
+
∞
)
M\colon \mathbb R^n \to [0,+\infty )
is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class
C
2
C^2
), and let
d
M
(
x
,
y
)
d^M(x,y)
be the (asymmetric) distance associated to
M
M
. Given an open domain
Ω
⊂
R
n
\Omega \subset \mathbb R^n
of class
C
2
C^2
, let
d
Ω
(
x
)
:=
inf
{
d
M
(
x
,
y
)
;
y
∈
∂
Ω
}
d_{\Omega }(x) := \inf \{d^M(x,y);\ y\in \partial \Omega \}
be the Minkowski distance of a point
x
∈
Ω
x\in \Omega
from the boundary of
Ω
\Omega
. We prove that a suitable extension of
d
Ω
d_{\Omega }
to
R
n
\mathbb R^n
(which plays the rôle of a signed Minkowski distance to
∂
Ω
\partial \Omega
) is of class
C
2
C^2
in a tubular neighborhood of
∂
Ω
\partial \Omega
, and that
d
Ω
d_{\Omega }
is of class
C
2
C^2
outside the cut locus of
∂
Ω
\partial \Omega
(that is, the closure of the set of points of nondifferentiability of
d
Ω
d_{\Omega }
in
Ω
\Omega
). In addition, we prove that the cut locus of
∂
Ω
\partial \Omega
has Lebesgue measure zero, and that
Ω
\Omega
can be decomposed, up to this set of vanishing measure, into geodesics starting from
∂
Ω
\partial \Omega
and going into
Ω
\Omega
along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point
x
∈
Ω
x\in \Omega
outside the cut locus the pair
(
p
(
x
)
,
d
Ω
(
x
)
)
(p(x), d_{\Omega }(x))
, where
p
(
x
)
p(x)
denotes the (unique) projection of
x
x
on
∂
Ω
\partial \Omega
, and we apply these techniques to the analysis of PDEs of Monge–Kantorovich type arising from problems in optimal transportation theory and shape optimization.