Let
M
M
be a compact
2
2
-manifold (without boundary)
C
1
{C^1}
-embedded in
R
3
{{\mathbf {R}}^3}
. Then there exists positive
σ
\sigma
such that, given any positive
τ
⩽
σ
\tau \leqslant \sigma
and any continuous map
f
:
M
→
R
f:M \to {\mathbf {R}}
, there exist points
p
p
,
q
q
,
r
∈
M
r \in M
, satisfying
‖
q
−
r
‖
=
‖
r
−
p
‖
=
‖
p
−
q
‖
=
τ
\left \| {q - r} \right \| = \left \| {r - p} \right \| = \left \| {p - q} \right \| = \tau
in the euclidean norm, for which
f
(
p
)
=
f
(
q
)
=
f
(
r
)
f(p) = f(q) = f(r)
.