We present precise anisotropic interpolation error estimates for smooth functions using a new geometric parameter and derive inverse inequalities on anisotropic meshes. In our theory, the interpolation error is bounded in terms of the diameter of a simplex and the geometric parameter. Imposing additional assumptions makes it possible to obtain anisotropic error estimates. This paper also includes corrections to an error in Theorem 2 of our previous paper, "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191).

翻译：我们用新的几何参数对平滑功能提出精确的反向内推误估计,并得出对反向不平等的反偏差。在我们的理论中,内推误以简单x和几何参数的直径为界限。加上其他假设,有可能获得对异向误差的估计。本文还包括对我们前一份文件“对异向内推误估计的一般理论”(日本工业和应用数学杂志,38 (2021) 163-191)的“理论2”中的一项误差的更正。