Minkowski chirality: a measure of reflectional asymmetry of convex bodies

Abstract

Using an optimal containment approach, we quantify the asymmetry of convex bodies in ${\mathbb{R}}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski chirality'' measures to Banach--Mazur distances and to each other, and prove their continuity with respect to the Hausdorff distance. In the planar case, we determine the reflection axes at which the Minkowski chirality of triangles and parallelograms is attained, and show that $\sqrt{2}$ is a tight upper bound on the chirality in both cases.