<s>
In	O
plane	O
geometry	O
the	O
Kovner	B-Algorithm
–	I-Algorithm
Besicovitch	I-Algorithm
measure	I-Algorithm
is	O
a	O
number	O
defined	O
for	O
any	O
bounded	O
convex	O
set	O
describing	O
how	O
close	O
to	O
being	O
centrally	B-Algorithm
symmetric	I-Algorithm
it	O
is	O
.	O
</s>
<s>
It	O
is	O
the	O
fraction	O
of	O
the	O
area	O
of	O
the	O
set	O
that	O
can	O
be	O
covered	O
by	O
its	O
largest	O
centrally	B-Algorithm
symmetric	I-Algorithm
subset	O
.	O
</s>
<s>
This	O
measure	O
is	O
one	O
for	O
a	O
set	O
that	O
is	O
centrally	B-Algorithm
symmetric	I-Algorithm
,	O
and	O
less	O
than	O
one	O
for	O
sets	O
whose	O
closure	O
is	O
not	O
centrally	B-Algorithm
symmetric	I-Algorithm
.	O
</s>
<s>
It	O
is	O
invariant	O
under	O
affine	B-Algorithm
transformations	I-Algorithm
of	O
the	O
plane	O
.	O
</s>
<s>
The	O
convex	O
sets	O
with	O
the	O
smallest	O
possible	O
Kovner	B-Algorithm
–	I-Algorithm
Besicovitch	I-Algorithm
measure	I-Algorithm
are	O
the	O
triangles	O
,	O
for	O
which	O
the	O
measure	O
is	O
2/3	O
.	O
</s>
<s>
The	O
curve	O
of	O
constant	O
width	O
with	O
the	O
smallest	O
possible	O
Kovner	B-Algorithm
–	I-Algorithm
Besicovitch	I-Algorithm
measure	I-Algorithm
is	O
the	O
Reuleaux	O
triangle	O
.	O
</s>
<s>
The	O
Kovner	B-Algorithm
–	I-Algorithm
Besicovitch	I-Algorithm
measure	I-Algorithm
of	O
any	O
given	O
convex	O
polygon	O
with	O
vertices	O
can	O
be	O
found	O
in	O
time	O
by	O
determining	O
a	O
translation	O
of	O
the	O
reflection	O
of	O
the	O
polygon	O
that	O
has	O
the	O
largest	O
possible	O
overlap	O
with	O
the	O
unreflected	O
polygon	O
.	O
</s>
<s>
Branko	O
Grünbaum	O
writes	O
that	O
the	O
Kovner	O
–	O
Besicovitch	O
theorem	O
was	O
first	O
published	O
in	O
Russian	O
,	O
in	O
a	O
1935	O
textbook	O
on	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
by	O
Mikhail	O
Lavrentyev	O
and	O
Lazar	O
Lyusternik	O
,	O
where	O
it	O
was	O
credited	O
to	O
Soviet	O
mathematician	O
and	O
geophysicist	O
.	O
</s>
<s>
Additional	O
proofs	O
were	O
given	O
by	O
Abram	O
Samoilovitch	O
Besicovitch	O
and	O
by	O
István	O
Fáry	O
,	O
who	O
also	O
proved	O
that	O
every	O
minimizer	O
of	O
the	O
Kovner	B-Algorithm
–	I-Algorithm
Besicovitch	I-Algorithm
measure	I-Algorithm
is	O
a	O
triangle	O
.	O
</s>
