Partially bounded sets of infinite width')

By Edward Thorp and Robert Whitley at Irvine

1 . Introduction

Clark [2] introduces these notions for a convex subset ^ of a real Banach space X: (1) A has finite width if A lies between two closed parallel hyperplanes, i. e. if there is a continuous linear functional :r* on X and two real numbers oc and ß with oc ^ x'^(x) ^ ß for all X in A^ {2) A has finite width in the direction a: Ф 0 if there is a constant w^ such that every line parallel to x intersects A in an interval of length no more than w^, i. e. if OCX -\- y and ßx -\- y belong to A then | ^ — oc\ ^ w^^ and (3) A is partially bounded (by K) if there is a finite least upper bound К for the radii of spheres contained in ^4. In what follows, we assume A is closed.

Clark shows that these conditions are equivalent when X is finite dimensional and asks whether this is true in an infinite dimensional Hubert space. (Ironically, the closure of the set W on page 615 of [2] furnishes an example of a closed convex body in separable Hilbert space which is partially bounded by 1 but which is of finite width in no direction.)

The implications (1) ==i> (2) =#> (3) are easy to establish. We show here that under various hypotheses, satisfied by many and perhaps by all infinite dimensional Banach spaces, none of the other possible implications hold. Specifically (1) there are partially bounded closed convex bodies of infinite width in any Banach space which has a separable infinite dimensional quotient, and (2) in any Banach space which has an infinite sional quotient having a Schauder basis, there is a closed convex set of infinite width but of finite width in some direction.

We note that Clark's discussion of the finite dimensional case is limited to convex bodies, i. e. convex sets with non-void interior. This is reasonable because a convex set with void interior is contained in a proper hyperplane, in the finite dimensional case. This shows that it trivially has all the properties of interest. It seemed to us possible but unlikely that Clark's interest was limited to convex bodies in the infinite dimensional case. However, we do give convex body examples whenever possible.

2 . PartiaUy Bounded Sets of Infinite Width

Theorem 1. Let X be an infinite dimensional Banach space which has a separable infinite dimensional quotient. Then X contains a closed convex set of infinite width which is partially bounded by 0.

^ ) This research was supported by the Air Force Office of Scientific Research under Grants AFOSR-AF- 111-63 and 70-1870 A and by the National Science Foundation under Grant GP-13288.