
280 Spatial Point Patterns: Methodology and Applications with R
Y is no n-empty and closed. Then fo r any location u, the empty- space distance is the minimum
distance from u to any point in Y,
d(u,Y) = min{ku −zk: z ∈Y}. (8.26)
See Figure 8.23. If u falls inside Y , then d(u,Y) = 0.
Figure 8.23. Distance from a location u to
the nearest point in a set Y (grey shading).
Now consider a random set Y, so that d(u,Y)
becomes a ra ndom variable. Assume that Y is a sta-
tionary random set (fo r details of these concepts, see
[473, 636].) The unconditional e m pty-space func-
tion or spherical contact distribution fu nction of Y
is
H
∗
(r) = P {d(u,Y) ≤ r}, r ≥