Given the increasing demand from wireless applications, designing
energy-efficient group communication protocols is of great importance to
multi-hop wireless networks. A group communication session involves a set of
member nodes, each of them needs to send a certain number of data packets to
all other members.
When data productions and consumptions are heavily unbalanced and when the
origins of data queries are spatially and temporally distributed, the so called
in-network data storage paradigm supersedes the conventional data collection
paradigm in wireless sensor networks (WSNs). In this paper, we first introduce
geometric quorum systems (along with their metrics) to incarnate the idea of
in-network data storage. These quorum systems are "geometric" because curves
(rather than discrete node sets) are used to form quorums.
In 1994, Martin Gardner stated a set of questions concerning the dissection
of a square or an equilateral triangle in three similar parts. Meanwhile,
Gardner's questions have been generalized and some of them are already solved.
In the present paper, we solve more of his questions and treat them in a much
more general context. Let $D\subset \mathbb{R}^d$ be a given set and let
$f_1,...,f_k$ be injective continuous mappings. Does there exist a set $X$ such
that $D = X \cup f_1(X) \cup ... \cup f_k(X)$ is satisfied with a
non-overlapping union?