Vol. 35, No. 1, 1970

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A characterization of the parallelepiped in En

Edwin Ogilvie Buchman and F. A. Valentine

Vol. 35 (1970), No. 1, 53–57

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.

Mathematical Subject Classification
Primary: 52.10
Received: 2 February 1970
Published: 1 October 1970
Edwin Ogilvie Buchman
F. A. Valentine