  In this paper we study the topological properties of wireless communication
maps and their usability in algorithmic design. We consider the SINR model,
which compares the received power of a signal at a receiver against the sum of
strengths of other interfering signals plus background noise. To describe the
behavior of a multi-station network, we use the convenient representation of a
\emph{reception map}. In the SINR model, the resulting \emph{SINR diagram}
partitions the plane into reception zones, one per station, and the
complementary region of the plane where no station can be heard. We consider
the general case where transmission energies are arbitrary (or non-uniform).
Under that setting, the reception zones are not necessarily convex or even
connected. This poses the algorithmic challenge of designing efficient point
location techniques as well as the theoretical challenge of understanding the
geometry of SINR diagrams. We achieve several results in both directions. We
establish a form of weaker convexity in the case where stations are aligned on
a line. In addition, one of our key results concerns the behavior of a
$(d+1)$-dimensional map. Specifically, although the $d$-dimensional map might
be highly fractured, drawing the map in one dimension higher "heals" the zones,
which become connected. In addition, as a step toward establishing a weaker
form of convexity for the $d$-dimensional map, we study the interference
function and show that it satisfies the maximum principle. Finally, we turn to
consider algorithmic applications, and propose a new variant of approximate
point location.
