Two different notions of approximate Birkhoff–James orthogonality in normed linear spaces have been introduced by Dragomir and Chmieliński. In the present paper we consider a global and a local approximate symmetry of the Birkhoff–James orthogonality related to each of the two definitions. We prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, we prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogonality is equivalent to some newly introduced geometric property. Our investigations complement and extend the scope of some recent results on a global approximate symmetry of the Birkhoff–James orthogonality.

Dragomir 和 Chmieliński 引入了赋范线性空间中近似 Birkhoff-James 正交性的两个不同概念。在本文中，我们考虑与两个定义中的每一个相关的 Birkhoff-James 正交性的全局和局部近似对称性。我们证明了所考虑的正交性在所有有限维 Banach 空间中的 Dragomir 意义上是近似对称的。对于另一种情况，我们证明对于有限维多面体 Banach 空间，正交性的近似对称性等价于一些新引入的几何性质。我们的研究补充并扩展了最近关于 Birkhoff-James 正交性的全局近似对称性的一些结果的范围。