Minkowski chirality: a measure of reflectional asymmetry of convex bodies

TL;DR

The paper develops a comprehensive framework to quantify reflectional asymmetry of convex bodies via Minkowski chiralities $\alpha_j(K)$, defined as the least dilation needed for containment of a reflected copy $\Phi_U(K)$ across $j$-dimensional subspaces $U$. It establishes sharp general bounds linking $\alpha_j(K)$ to the classical Minkowski asymmetry $\alpha_0(K)$ and the Banach–Mazur distance to the Euclidean ball, with a unified bound $\alpha_j(K) \leq \sqrt{\alpha_0(K)n}$; continuity in the Hausdorff metric and the existence of optimal subspaces are proven. In the planar case, the authors provide complete characterizations: $\alpha_1(K) \in [1,\sqrt{2})$ for triangles and $\alpha_1(K) \in [1,\sqrt{2}]$ for parallelograms, with explicit formulas and phase diagrams identifying which reflection axis attains the bound. The results connect to John ellipsoids and Banach–Mazur distances, yielding a robust toolbox for comparing asymmetry to simplices and guiding further study of higher-dimensional and special-body cases.

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.

Figures (10)

• Figure 1: Case 2 (left) and Case 3 (right) in the proof of \ref{['thm:cool-proof']}: Parallelogram $K$ (green), reflection axis $U$ (grey), $\Phi_U(K)$ (red), and $C=R(K,\Phi_U(K))\Phi_U(K)$ (black).
• Figure 2: Case 4 in the proof of \ref{['thm:cool-proof']}: Let all edges of $A(K)$ (green) contain a vertex of a parallelogram $P$ (red, dotted) that is not a square. Then $P$ cannot be a dilatation of $A(\Phi_U(K))$ since not all edges of $R(A(K),P)P$ (red, dashed) contain a vertex of $A(K)$ in their relative interiors (right panel). If $U$ is a principal axis of the John ellipsoid, $A(K)$ (green) and $A(\Phi_U(K))$ (red) are squares, both with John ellipsoid $\mathds{B}_2^2$ and vertices on $\mathop{\mathrm{bd}}\nolimits(\sqrt{2}\mathds{B}_2^2)$. In the relative interior of each edge of $R(K,\Phi_U(K)) A(\Phi_U(K))$, there exists a vertex of $A(K)$ (left panel).
• Figure 3: Optimal containment of a parallelogram $K$ (green) in the dilated mirror image after reflection across the reflection axis $U$ from the proof of \ref{['prop:angle-bisector-parallelogram']} for the case of a bisector of an interior angle (left panel) and a bisector of the diagonals (right panel): reflection axis $U$ (grey), $\Phi_U(K)$ (red), and an appropriate translate of $R(K,\Phi_U(K))\Phi_U(K)$ (black).
• Figure 4: The obtuse angle formed by the diagonals of the parallelogram is denoted by $\delta$.
• Figure 5: We parametrize nontrivial parallelograms $K$ in three different ways by the larger interior angle $\theta\in (\frac{\uppi}{2},\uppi)$, the side length ratio $r\in (1,\infty)$, the larger angle $\delta\in (\frac{\uppi}{2},\uppi)$ formed by the diagonals, and the coordinates $x,y\in\mathds{R}$. (In the latter case, we assume that the vertices of $K$ are $\pm (1,0)$ and $\pm (x,y)$, where $x^2+y^2<1$ and $x,y > 0$.) The solid lines separate regions which correspond to the reflection axis at which the Minkowski chirality $\alpha_1$ is attained: a bisector of an interior angle for $\mathcal{B}$, an angle bisector of the diagonals for $\mathcal{D}$, and a principal axis of the John ellipse for $\mathcal{J}$. The dashed line indicates the parallelograms with $\alpha_1(K) = \sqrt{2}$.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3