Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T07:27:39Z http://mathoverflow.net/feeds/question/60426 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60426/lipschitz-homeomorphisms-between-spheres-of-n-dimensional-spaces Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces Bill Johnson 2011-04-03T11:27:19Z 2011-04-03T11:27:19Z <p>Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm.</p> <p><strong>Question:</strong> Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so that the Lipschitz constant of $f_N$ is $C$. Is $C_N$ bounded as $N\to \infty$?</p> <p>Notice that the $\ell_\infty^N$ norm is $2$-equivalent to the $\ell_p^N$ norm with $p = \log_2 N$, and the Mazur map from $B_p^N$ to $B_2^N$ has Lipschitz constant about $p$, so $C_N$ grows no faster than $\log N$. </p> <p>(The Mazur map from $\ell_p$ to $\ell_2$ is defined for unit vectors to be $x\mapsto |x|^{p/2}\rm{sign}\ x$ and extended to be positively homogeneous on $\ell_p$.)</p>