Banach-Mazur distance

The Banach-Mazur distance measures how "similar" two shapes (specifically, convex bodies) are in terms of shape and size. It looks at the smallest possible scaling and stretching needed to transform one shape into another so that they fit closely together. A smaller distance means the shapes are more similar, essentially like being deformable into each other with minimal distortion. This concept helps mathematicians understand the geometry and complexity of different shapes in spaces that extend beyond simple objects, providing a way to compare their structure quantitatively.