I am interested in the smallest coordinate of a vector. Given a positive real vector $x=(x_1,...,x_n)$, I need your help to determine if the function $f(x)= min_{1\leq i \leq n}x_i$ is Lipschitz. If not, can we for example restrict $f$ on some subspace of $\mathbb{R}^n$ so that we get Lipschitzity? (Eg say $0\leq x_i \leq M$ for all the coordinates).
I need $f$ to be Lipschitz because I am currently working on an ODE from a competition model where every step is led but the smallest individuals. Ideally, this would lead to uniqueness of the solution for the model. I am also open to any lead.
Thanks!
One has that$$|f(x)-f(y)| = |\min_{1\le i \le N} x_i - \min_{1\le i \le N} y_i| \overset{(\star)}\le \max_{1\le i \le N} |x_i-y_i| = \|x-y\|_\infty.$$
As all norms are equivalent on a finite dimensional space, there is a constant $C>0$ such that $||x-y||_\infty\leq C ||x-y||$ for all $x,y\in\mathbb{R}^n$ (where $||\cdot ||$ represents the standard Euclidean norm).
From here, $f$ is Lipschitz with constant $C$.
To prove ($\star$), assume WLOG that $\displaystyle \min_{1\le i \le N} x_i \geq \min_{1\le i \le N} y_i$. Then, if $j$ is the index such that $y_j=\displaystyle \min_{1\le i \le N} y_i$,$$|\min_{1\le i \le N} x_i - \min_{1\le i \le N} y_i|=\min_{1\le i \le N} x_i - \min_{1\le i \le N} y_i= \min_{1\le i \le N} x_i -y_j\leq x_j-y_j\\\leq |x_j-y_j|\leq \max_{1\le i \le N} |x_i-y_i|$$