It is a well-known that if $f$ is $L$-lipschitz continuous, for any $x, y$, $$ |f(x)- f(y)| \le L|x-y|.$$ I am wondering if there is any notion for the reverse direction, i.e, for any $x,y$, $$ \ell |x - y| \le |f(x) - f(y)|. $$ and if it is somehow known concept, is there any name for this? (e.g. reverse lipschitz?, (i am just guessing)).
Any comments or suggestions will be very appreciated. Thanks.
Lipschitz maps with $L<1$ are called contractive (sometimes they are also called that if $L=1$ depending on what you're doing). This is because repeatedly applying them contracts distances between points. Analogously, a "reverse Lipschitz" map in your sense with $\ell>1$ is called expansive. I doubt there is a term for a "reverse Lipschitz" map with no restrictions on $\ell$.