<s>
In	O
mathematics	O
,	O
the	O
lower	B-Algorithm
envelope	I-Algorithm
or	O
pointwise	B-Algorithm
minimum	I-Algorithm
of	O
a	O
finite	O
set	O
of	O
functions	O
is	O
the	O
pointwise	B-Algorithm
minimum	I-Algorithm
of	O
the	O
functions	O
,	O
the	O
function	O
whose	O
value	O
at	O
every	O
point	O
is	O
the	O
minimum	O
of	O
the	O
values	O
of	O
the	O
functions	O
in	O
the	O
given	O
set	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
lower	B-Algorithm
envelope	I-Algorithm
can	O
also	O
be	O
extended	O
to	O
partial	B-Algorithm
functions	I-Algorithm
by	O
taking	O
the	O
minimum	O
only	O
among	O
functions	O
that	O
have	O
values	O
at	O
the	O
point	O
.	O
</s>
<s>
The	O
upper	B-Algorithm
envelope	I-Algorithm
or	O
pointwise	B-Algorithm
maximum	I-Algorithm
is	O
defined	O
symmetrically	O
.	O
</s>
<s>
For	O
continuous	O
functions	O
from	O
a	O
given	O
class	O
,	O
the	O
lower	O
or	O
upper	B-Algorithm
envelope	I-Algorithm
is	O
a	O
piecewise	B-Algorithm
function	I-Algorithm
whose	O
pieces	O
are	O
from	O
the	O
same	O
class	O
.	O
</s>
<s>
For	O
functions	O
of	O
a	O
single	O
real	O
variable	O
whose	O
graphs	O
have	O
a	O
bounded	O
number	O
of	O
intersection	O
points	O
,	O
the	O
complexity	O
of	O
the	O
lower	O
or	O
upper	B-Algorithm
envelope	I-Algorithm
can	O
be	O
bounded	O
using	O
Davenport	O
–	O
Schinzel	O
sequences	O
,	O
and	O
these	O
envelopes	O
can	O
be	O
computed	O
efficiently	O
by	O
a	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
that	O
computes	O
and	O
then	O
merges	O
the	O
envelopes	O
of	O
subsets	O
of	O
the	O
functions	O
.	O
</s>
<s>
For	O
convex	O
functions	O
or	O
quasiconvex	O
functions	O
,	O
the	O
upper	B-Algorithm
envelope	I-Algorithm
is	O
again	O
convex	O
or	O
quasiconvex	O
.	O
</s>
<s>
The	O
lower	B-Algorithm
envelope	I-Algorithm
is	O
not	O
,	O
but	O
can	O
be	O
replaced	O
by	O
the	O
lower	O
convex	O
envelope	O
to	O
obtain	O
an	O
operation	O
analogous	O
to	O
the	O
lower	B-Algorithm
envelope	I-Algorithm
that	O
maintains	O
convexity	O
.	O
</s>
<s>
The	O
upper	O
and	O
lower	B-Algorithm
envelopes	I-Algorithm
of	O
Lipschitz	O
functions	O
preserve	O
the	O
property	O
of	O
being	O
Lipschitz	O
.	O
</s>
<s>
However	O
,	O
the	O
lower	O
and	O
upper	B-Algorithm
envelope	I-Algorithm
operations	O
do	O
not	O
necessarily	O
preserve	O
the	O
property	O
of	O
being	O
a	O
continuous	O
function	O
.	O
</s>
