A bounded open set in the
Euclidean plane E2 which has a parallelogram as its boundary will be called a two
dimensional open parallelepiped. The n-dimensional analogue of such an object is
called an open parallelepiped in En. In order to motivate the characterization
presented here, it is desirable to recall the characterization of starshaped sets given
by Krasnosel’skií. In 1946 Krasnosel’skii proved that a bounded closed set S in En
is starshaped if and only if each set of s + 1 points of S with s ≦ n can
see at least one point of lS via S. As is well known, this result fails if S is
unbounded. However, under what circumstances can a set S see infinity via S in
the same direction? This and related questions led to the following which
is first stated here intuitively. The open parallelepiped in En is the only
nonempty bounded open convex set ∕S in En which has the property that every
2n − 1 of its boundary points can see infinity in a same direction via the
complement of S. This result is related to a theorem of Steinitz on convex
hulls.
