This paper defines the notion of a best simultaneous Diophantine approximation to a vector
α
\alpha
in
R
n
R^n
with respect to a norm
‖
⋅
‖
\left \| \,\cdot \, \right \|
on
R
n
R^n
. Suppose
α
\alpha
is not rational and order the best approximations to
α
\alpha
with respect to
‖
⋅
‖
\left \|\, \cdot \, \right \|
by increasing denominators
1
=
q
1
>
q
2
>
⋯
1=q_1 > q_2 > \cdots
. It is shown that these denominators grow at least at the rate of a geometric series, in the sense that
\[
g
(
α
,
‖
⋅
‖
)
=
lim inf
k
→
∞
(
q
k
)
1
/
k
≥
1
+
1
2
n
+
1
g\left ( {\alpha ,\,\left \| {\,\cdot \,} \right \|} \right ) = \liminf \limits _{k \to \infty } {({q_k})^{1/k}} \geq 1 + \frac {1}{{{2^{n + 1}}}}
\]
. Let
g
(
‖
⋅
‖
)
g\left ( {\left \|\, \cdot \, \right \|} \right )
denote the infimum of
g
(
α
,
‖
⋅
‖
)
g\left ( {\alpha ,\,\left \| {\,\cdot \,} \right \|} \right )
over all
α
\alpha
in
R
n
R^n
with an irrational coordinate. For the sup norm
‖
⋅
‖
s
\left \|\, \cdot \,\right \|_s
on
R
2
R^2
it is shown that
g
(
‖
⋅
‖
s
)
≥
θ
=
1.270
+
g\left ( {\left \| \, \cdot \, \right \|}_s \right )\ge \theta =1.270^{+}
where
θ
4
=
θ
2
+
1
\theta ^4=\theta ^{2}+1
.