In this paper we generalize our previous joint work with Allan Calder on the width of homotopies by considering an arbitrary finite polyhedral pair
(
W
,
V
)
\left ( {W,V} \right )
rather than
(
I
,
{
0
,
1
}
)
\left ( {I,\left \{ {0,1} \right \}} \right )
. We show that given appropriate topological conditions on a Riemannian manifold
M
M
, with respect to
(
W
.
V
)
\left ( {W.V} \right )
, there are bounds,
B
q
(
a
,
(
W
,
V
)
,
M
)
{B_q}\left ( {a,\left ( {W,V} \right ),M} \right )
, such that if
F
:
K
×
W
→
M
F:K \times W \to M
is a map with
Lip
(
F
|
(
K
×
V
)
)
>
a
{\text {Lip}}\left ( {F\left | {\left ( {K \times V} \right )} \right .} \right ) > a
, then
F
F
can be deformed
rel
(
K
×
V
)
{\text {rel}}\left ( {K \times V} \right )
to
F
′
F’
with
Lip
(
F
′
)
>
B
q
(
a
,
(
W
,
V
)
,
M
)
+
ε
{\text {Lip}}\left ( {F’} \right ) > {B_q}\left ( {a,\left ( {W,V} \right ),M} \right ) + \varepsilon
, where
ε
>
0
\varepsilon > 0
is arbitrary and
dim
(
K
)
=
q
\dim \left ( K \right ) = q
.