Relating spans of some continua homeomorphic to $S^ n$

Abstract:

It has been conjectured that ${\sigma ^ * }(X) \geq \tfrac {1}{2}\sigma _0^ * (X)$ for each nonempty connected metric space $X$. In this paper we show that ${\sigma ^ * }(X) \geq (\sqrt 3 /2)\sigma _0^ * (X)$ when $X \subset {R^n}$ is homeomorphic to ${S^{n - 1}}$ for $n = 2,3, \ldots$ 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 $\sqrt 3 /2$. It has also been conjectured that ${\sigma ^ * }(X) \geq \sigma (X)/2$ and that $\sigma _0^ * (X) \geq {\sigma _0}(X)/2$ for each nonempty connected metric space $X$. We show that these two inequalities hold when $X \subset {R^n}$ is homeomorphic to ${S^{n - 1}}$ for $n = 3,4, \ldots$ and $A$ is convex where $A$ is the bounded component of ${R^n} - X$.