Glasnik Matematicki, Vol. 42, No.2 (2007), 411-425.
THE SPANS OF FIVE STAR-LIKE SIMPLE CLOSED CURVES
Department of Mathematics, University of Louisiana at Lafayette,
LA 70504-1010, USA
Abstract. Let X be a continuum, that is a compact, connected, nonempty
metric space. The span of X is the least upper bound of the set of
real numbers r which satisfy the following conditions: there
exists a continuum, C, contained in
X × X such that d(x,y)
is larger than or equal to r for all (x,y) in C
and p1(C) = p2(C),
where p1, p2 are the usual projection maps. The
following question has been asked. If X and Y are two simple
closed curves in the plane and Y is contained in the bounded
component of the plane minus X, then is the span of X larger
than the span of Y? We define a set of simple closed curves, which
we refer to as being five star-like. We answer this question in the
affirmative when X is one of these simple closed curves. We
calculate the spans of the simple closed curves in this collection
and consider the spans of various geometric objects related to these
simple closed curves.
2000 Mathematics Subject Classification.
Key words and phrases. Span, simple closed curve.
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