Faried Nashat, Morsy Ahmed, Hussein Aya M
1Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt.
2Department of Mathematics, Faculty of Arts and Sciences at Wadi Addawasir, Prince Sattam Bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia.
J Inequal Appl. 2018;2018(1):31. doi: 10.1186/s13660-018-1624-z. Epub 2018 Feb 5.
In 1960, Dvoretzky proved that in any infinite dimensional Banach space and for any [Formula: see text] there exists a subspace of of arbitrary large dimension -iometric to Euclidean space. A main tool in proving this deep result was some results concerning asphericity of convex bodies. In this work, we introduce a simple technique and rigorous formulas to facilitate calculating the asphericity for each set that has a nonempty boundary set with respect to the flat space generated by it. We also give a formula to determine the center and the radius of the smallest ball containing a nonempty nonsingleton set in a linear normed space, and the center and the radius of the largest ball contained in it provided that has a nonempty boundary set with respect to the flat space generated by it. As an application we give lower and upper estimations for the asphericity of infinite and finite cross products of these sets in certain spaces, respectively.
1960年，德沃雷茨基证明，在任何无限维巴拿赫空间中，对于任意的[公式：见正文]，都存在一个维度任意大的子空间，它与欧几里得空间等距。证明这一深刻结果的一个主要工具是一些关于凸体非球面性的结果。在这项工作中，我们引入了一种简单的技术和严格的公式，以便于计算每个相对于由其生成的平坦空间具有非空边界集的集合的非球面性。我们还给出了一个公式，用于确定线性赋范空间中包含非空非单点集的最小球的中心和半径，以及假设该集合相对于由其生成的平坦空间具有非空边界集时，包含在其中的最大球的中心和半径。作为应用，我们分别给出了这些集合在某些空间中的无限和有限叉积的非球面性的下界和上界估计。
