We prove that the trace of the space
C
1
,
ω
(
R
n
)
C^{1,\omega }({\mathbb R}^n)
to an arbitrary closed subset
X
⊂
R
n
X\subset {\mathbb R}^n
is characterized by the following “finiteness” property. A function
f
:
X
→
R
f:X\rightarrow {\mathbb R}
belongs to the trace space if and only if the restriction
f
|
Y
f|_Y
to an arbitrary subset
Y
⊂
X
Y\subset X
consisting of at most
3
⋅
2
n
−
1
3\cdot 2^{n-1}
can be extended to a function
f
Y
∈
C
1
,
ω
(
R
n
)
f_Y\in C^{1,\omega }({\mathbb R}^n)
such that
\[
sup
{
‖
f
Y
‖
C
1
,
ω
:
Y
⊂
X
,
card
Y
≤
3
⋅
2
n
−
1
}
>
∞
.
\sup \{\|f_Y\|_{C^{1,\omega }}:~Y\subset X, ~\operatorname {card} Y\le 3\cdot 2^{n-1}\}>\infty .
\]
The constant
3
⋅
2
n
−
1
3\cdot 2^{n-1}
is sharp.
The proof is based on a Lipschitz selection result which is interesting in its own right.