Banach-Mazur Distances

The Banach-Mazur distance is a way mathematicians measure how similar two shapes, specifically convex bodies or Banach spaces, are when scaled, rotated, and stretched. Think of it as a "closeness" score: if the distance is 1, they are essentially the same shape; larger values indicate more difference. It considers the best possible way to fit one shape inside another using linear transformations and scaling. This concept helps understand the structural relationships between different mathematical spaces and shapes, providing a way to compare their geometric properties precisely.