RELATING SPANS OF SOME CONTINUA HOMEOMORPHIC TO S-N

It has been conjectured that sigma* (X) greater-than-or-equal-to 1/2-sigma-0* (X) for each nonempty connected metric space X. In this paper we show that sigma* (X) greater-than-or-equal-to (square-root 3/2)sigma-0* (X) when X subset-of R(n) is homeomorphic to S(n-1) for n = 2, 3,... and A is convex where A is the bounded component of R(n) - X. We also show that under certain conditions a lower bound for the ratio sigma* (X)/sigma-0* (X) is larger than square-root 3/2. It has also been conjectured that sigma* (X) greater-than-or-equal-to sigma(X)/2 and that sigma-0* (X) greater-than-or-equal-to sigma-0(X)/2 for each nonempty connected metric space X. We show that these two inequalities hold when X subset-of R(n) is homeomorphic to S(n-1) for n = 3, 4,... and A is convex where A is the bounded component of R(n) - X.