  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).
