The centroid Banach-Mazur distance between the parallelogram and the triangle

TL;DR

This paper introduces a centroid-based Banach-Mazur distance and calculates it explicitly for the case of a parallelogram and a triangle in two-dimensional space, revealing a specific value of 2.5.

Contribution

The paper defines a new centroid-constrained Banach-Mazur distance and computes its exact value for parallelogram and triangle pairs in the plane.

Findings

Centroid Banach-Mazur distance between parallelogram and triangle is 2.5.

The new distance measure incorporates centroid alignment constraints.

Explicit calculation for specific convex bodies in Euclidean space.

Abstract

Let $C$ and $D$ be convex bodies in the Euclidean space $E^d$. We define the centroid Banach-Mazur distance $\delta_{BM}^{\rm cen} (C, D)$ similarly to the classic Banach-Mazur distance $\delta_{BM} (C, D)$, but with the extra requirement that the centroids of $C$ and an affine image of $D$ coincide. We prove that for the parallelogram $P$ and the triangle $T$ in $E^2$ we have $\delta_{BM}^{\rm cen} (P, T) = \frac{5}{2}$.