
Priors on Hypergraphical Models via Simplicial Complexes 401
Proposition 19.4.1. Every feasible graph in R
d
may be represented in the
form G(V, r, A) for a collection V of n points in the unit ball B
d
and for r =
1
n
.
Proof
Let V = {V
1
, . . . , V
n
} ⊂ B
d
be a set of points and r > 0 a radius such that
G(V, r, Čech) is the empty graph. Then the balls V
i
+rB
d
are disjoint and their
union with d-dimensional volume nω
d
r
d
lies wholly within the ball (1 + r)B
d
of volume ω
d
(1 + r)
d
(where ω
d
= π
d/2
/Γ(1+d/2) is the volume of the unit
ball), so n < (1 +
1
r
)
d
.
Slightly stronger, the empty graph may not be attained as G(V, r, Čech)
for any r ≥ 1/[(n/p
d
)
1/d
− 1] where p
d
is ...