Timeline for $\varepsilon$-covering number after diffeomorphism

7 events

when toggle format	what		by	license	comment
Jan 18, 2014 at 16:17	vote	accept	Ayman Moussa
Jan 17, 2014 at 15:01	answer	added	Vidit Nanda		timeline score: 1
Jan 17, 2014 at 11:06	comment	added	Vidit Nanda		Yes. And note also that $\phi$ must have a $\mathscr{C}^1$ inverse by assumption with a Lipschitz constant of its own, and so you can always map the covering along this inverse to get a similar inequality bounding $N(\epsilon,\phi(K))$ from the other side.
Jan 17, 2014 at 9:06	comment	added	Ayman Moussa		Okay, to be sure to understand, what you're saying is that if $C$ is the Lipschitz constant of $\phi$, we have always $N(\varepsilon,\varphi(K)) \leq N(\frac{\varepsilon}{C},K)$, just by transfering the covering, right ?
Jan 17, 2014 at 3:20	comment	added	Vidit Nanda		Fix $\epsilon$. Since $\phi$ is $\mathscr{C}^1$ it is Lipschitz on compact sets. Consider the compact set $K^\epsilon$ which equals the union of closed $\epsilon$-balls around points in $K$. This quickly gives you an upper bound on $N(\epsilon,\phi(K))$ in terms of the Lipschitz constant. In order to get an analogous lower bound, you would probably require bi-Lipschitzness of $\phi$.
Jan 16, 2014 at 22:12	answer	added	Ken Richardson		timeline score: 2
Jan 16, 2014 at 18:29	history	asked	Ayman Moussa	CC BY-SA 3.0