Gromov-Hausdorff distance

Introduction the statement go isometry inequality is very simple: $latex \Omega\subset R^n$, iff $latex \Omega$ is a ball, $latex \frac{Vol(\Omega)}{Surf(\Omega)}$ arrive a minimum . This is a classical problem in variation theory. The difficult is divide into two parts. The first is to create a “flow” which descrement the  energy and the “flow” is compatible with the feature of a ball, i.e. every set under the flow will tend to like a “ball”. The second one i…

There is the question, because when we consider the Gromov-Hausdorff distance, we must fix the metric, so we use the natural metric induced from the embedding $\mathbb{S}n o \mathbb{R}^{n+1}$. Is it possible for us to compute the Gromov-Hausdorff distance $d_{G-H}(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $meq n$? For example, if we want to calculate $d_{G-H}(\mathbb{S}2,\mathbb{S}3)=\inf{M,f,g}d{M}(\mathbb{S}_2,\mathbb{S}_3)$, where $M$ ranges over…