Extension of lipschitz functions along a curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:42:22Z http://mathoverflow.net/feeds/question/106388 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106388/extension-of-lipschitz-functions-along-a-curve Extension of lipschitz functions along a curve warsaga 2012-09-04T22:51:44Z 2012-09-06T14:38:13Z <p>Given a curve $\gamma$ in a Banach space $X$ and a function f defined along the curve s.t. $$\big\Vert f(\gamma(t))-f(\gamma(s))\big\Vert\leq L\big\Vert\gamma(t)-\gamma(s)\big\Vert$$ is it possible to extend the Lipschitz functions to the whole of $X$? </p> http://mathoverflow.net/questions/106388/extension-of-lipschitz-functions-along-a-curve/106402#106402 Answer by jbc for Extension of lipschitz functions along a curve jbc 2012-09-05T05:30:25Z 2012-09-05T05:30:25Z <p>The basic extension result for Lipschitz functions is the theorem of Kirszbaum. This works for functions with values in $\mathbb{R}^n$ and is expounded in Federer's book on Geometric Measure Theory. I think that it even works for functions with values in Hilbert space but can't trace a reference.</p> http://mathoverflow.net/questions/106388/extension-of-lipschitz-functions-along-a-curve/106405#106405 Answer by Pietro Majer for Extension of lipschitz functions along a curve Pietro Majer 2012-09-05T06:41:43Z 2012-09-05T06:41:43Z <p>If you mean a real-valued function $f$, yes, and keeping the same constant $L$, by a simple construction. Check the last mentioned property listed <a href="http://en.wikipedia.org/wiki/Lipschitz_function#Properties" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/106388/extension-of-lipschitz-functions-along-a-curve/106459#106459 Answer by Bill Johnson for Extension of lipschitz functions along a curve Bill Johnson 2012-09-05T19:50:49Z 2012-09-05T19:50:49Z <p>It is not always possible to extend when $X$ is a Banach space. Take a Banach space $Y_n$ which contains an $n$ dimensional subspace $E_n$ such that every projection from $Y_n$ onto $E_n$ has norm at least $C_n$ with $C_n\to \infty$. ($Y_n$ can e.g. be $L_1$ and $E_n$ the span of $n$ IID gaussian random variables; then $C_n$ is of order $n^{1/2}$.) Let $X_n = Y_n \oplus_2 E_n$. For the curve in $Y_n$ take any curve in the unit sphere of $E_n \oplus {0}$ that contains an $\epsilon_n$ net $A_n$ of the unit sphere of $E_n \oplus {0}$. For $f_n$ take the natural isometry from $E_n \oplus {0}$ onto ${0} \oplus E_n$ restricted to the curve. Let $F_n$ be an extension of $f_n$ to a Lipschitz mapping on $X_n$; WLOG $F_n$ maps into ${0} \oplus E_n$ since this is a norm one complemented subspace of $X_n$. Let $G_n$ be the positively homogeneous extension of the restriction of $F_n$ to the unit sphere of $X_n$. Then the Lipschitz constant of $G_n$ is at most three times the Lipschitz constant of $F_n$. Compose $G_n$ with the obvious isometry from ${0} \oplus E_n$ onto $E_n \oplus {0}$. The restriction of this map to $Y_n$ gives a positively homogenous mapping from $Y_n$ into $E_n$ that is the identity on $A_N$. By the arguments in  Johnson, William B.(1-OHSN); Lindenstrauss, Joram(IL-HEBR) Extensions of Lipschitz mappings into a Hilbert space. Conference in modern analysis and probability (New Haven, Conn., 1982), 189–206, Contemp. Math., 26, Amer. Math. Soc., Providence, RI, 1984 </p> <p>we conclude that if $\epsilon_n$ is sufficiently small, there is a projection from $Y_n$ onto $E_n$ whose norm is no worse than something like ten times the Lipschitz constant of $G_n$. </p> <p>All of this shows that you cannot get Lipschitz extensions with controlled norms. Take an infinite direct sum to get an example where you cannot get any Lipschitz extension.</p>