<s>
In	O
geometry	O
,	O
an	O
Hadamard	B-Algorithm
space	I-Algorithm
,	O
named	O
after	O
Jacques	O
Hadamard	O
,	O
is	O
a	O
non-linear	O
generalization	O
of	O
a	O
Hilbert	O
space	O
.	O
</s>
<s>
In	O
a	O
Hilbert	O
space	O
,	O
the	O
above	O
inequality	O
is	O
equality	O
(	O
with	O
)	O
,	O
and	O
in	O
general	O
an	O
Hadamard	B-Algorithm
space	I-Algorithm
is	O
said	O
to	O
be	O
if	O
the	O
above	O
inequality	O
is	O
equality	O
.	O
</s>
<s>
A	O
flat	O
Hadamard	B-Algorithm
space	I-Algorithm
is	O
isomorphic	O
to	O
a	O
closed	O
convex	O
subset	O
of	O
a	O
Hilbert	O
space	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
normed	O
space	O
is	O
an	O
Hadamard	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
a	O
Hilbert	O
space	O
.	O
</s>
<s>
The	O
geometry	O
of	O
Hadamard	B-Algorithm
spaces	I-Algorithm
resembles	O
that	O
of	O
Hilbert	O
spaces	O
,	O
making	O
it	O
a	O
natural	O
setting	O
for	O
the	O
study	O
of	O
rigidity	O
theorems	O
.	O
</s>
<s>
In	O
a	O
Hadamard	B-Algorithm
space	I-Algorithm
,	O
any	O
two	O
points	O
can	O
be	O
joined	O
by	O
a	O
unique	O
geodesic	O
between	O
them	O
;	O
in	O
particular	O
,	O
it	O
is	O
contractible	O
.	O
</s>
<s>
Quite	O
generally	O
,	O
if	O
is	O
a	O
bounded	O
subset	O
of	O
a	O
metric	O
space	O
,	O
then	O
the	O
center	O
of	O
the	O
closed	O
ball	O
of	O
the	O
minimum	O
radius	O
containing	O
it	O
is	O
called	O
the	O
circumcenter	O
of	O
Every	O
bounded	O
subset	O
of	O
a	O
Hadamard	B-Algorithm
space	I-Algorithm
is	O
contained	O
in	O
the	O
smallest	O
closed	O
ball	O
(	O
which	O
is	O
the	O
same	O
as	O
the	O
closure	O
of	O
its	O
convex	O
hull	O
)	O
.	O
</s>
<s>
If	O
is	O
the	O
group	O
of	O
isometries	O
of	O
a	O
Hadamard	B-Algorithm
space	I-Algorithm
leaving	O
invariant	O
then	O
fixes	O
the	O
circumcenter	O
of	O
(	O
Bruhat	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
fixed	I-Algorithm
point	I-Algorithm
theorem	I-Algorithm
)	O
.	O
</s>
<s>
The	O
analog	O
holds	O
for	O
a	O
Hadamard	B-Algorithm
space	I-Algorithm
:	O
a	O
complete	O
,	O
connected	O
metric	O
space	O
which	O
is	O
locally	O
isometric	O
to	O
a	O
Hadamard	B-Algorithm
space	I-Algorithm
has	O
an	O
Hadamard	B-Algorithm
space	I-Algorithm
as	O
its	O
universal	O
cover	O
.	O
</s>
<s>
Examples	O
of	O
Hadamard	B-Algorithm
spaces	I-Algorithm
are	O
Hilbert	O
spaces	O
,	O
the	O
Poincaré	O
disc	O
,	O
complete	O
real	O
trees	O
(	O
for	O
example	O
,	O
complete	O
Bruhat	O
–	O
Tits	O
building	O
)	O
,	O
-space	O
with	O
and	O
and	O
Hadamard	B-Algorithm
manifolds	I-Algorithm
,	O
that	O
is	O
,	O
complete	O
simply-connected	O
Riemannian	B-Architecture
manifolds	I-Architecture
of	O
nonpositive	O
sectional	O
curvature	O
.	O
</s>
<s>
Important	O
examples	O
of	O
Hadamard	B-Algorithm
manifolds	I-Algorithm
are	O
simply	O
connected	O
nonpositively	O
curved	O
symmetric	O
spaces	O
.	O
</s>
<s>
Applications	O
of	O
Hadamard	B-Algorithm
spaces	I-Algorithm
are	O
not	O
restricted	O
to	O
geometry	O
.	O
</s>
<s>
The	O
solution	O
begins	O
by	O
constructing	O
a	O
configuration	O
space	O
for	O
the	O
dynamical	O
system	O
,	O
obtained	O
by	O
joining	O
together	O
copies	O
of	O
corresponding	O
billiard	O
table	O
,	O
which	O
turns	O
out	O
to	O
be	O
an	O
Hadamard	B-Algorithm
space	I-Algorithm
.	O
</s>
