| Abstrakt: |
It has been conjectured that $ {\sigma ^ * }(X) \geq \tfrac{1}{2}\sigma _0^ * (X)$. In this paper we show that $ {\sigma ^ * }(X) \geq (\sqrt 3 /2)\sigma _0^ * (X)$ $ X \subset {R^n}$ $ {S^{n - 1}}$ $ n = 2,3, \ldots $ is convex where $ A$. We also show that under certain conditions a lower bound for the ratio $ {\sigma ^ * }(X)/\sigma _0^ * (X)$ $ \sqrt 3 /2$ $ {\sigma ^ * }(X) \geq \sigma (X)/2$ $ \sigma _0^ * (X) \geq {\sigma _0}(X)/2$. We show that these two inequalities hold when $ X \subset {R^n}$ $ {S^{n - 1}}$ $ n = 3,4, \ldots $ is convex where $ A$. |
