The relationships of spans of convex continua in 𝑅ⁿ

Abstract

It has been conjectured that σ ∗ ( X ) ≥ 1 2 σ 0 ∗ ( X ) {\sigma ^ * }(X) \geq \tfrac {1}{2}\sigma _0^ * (X) for each nonempty connected metric space X X . In this paper we show that σ ∗ ( X ) ≥ 13 4 σ 0 ∗ ( X ) {\sigma ^ * }(X) \geq \tfrac {{\sqrt {13} }}{4}\sigma _0^ * (X) for each convex continuum X X in R n {R^n} . We also show that under certain conditions a lower bound for the ratio σ ∗ ( X ) / σ 0 ∗ ( X ) {\sigma ^ * }(X)/\sigma _0^ * (X) is larger than 13 4 \tfrac {{\sqrt {13} }}{4} . It has also been conjectured that σ ∗ ( X ) ≥ σ ( X ) / 2 {\sigma ^ * }(X) \geq \sigma (X)/2 and that σ 0 ∗ ( X ) ≥ σ 0 ( X ) / 2 \sigma _0^ * (X) \geq {\sigma _0}(X)/2 for each nonempty connected metric space X X . We show that these two inequalities hold when X X is a convex continuum in R n {R^n} .