What type of growth conditions can establish \$||f(x)||\geq K \text{dist}(x,\mathcal{X})\$?

[Vulture1991]

Hello! I have the following question:

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$. Is there any class of class of functions and some kind of "growth conditions" such that bounds like below can be established:
\begin{equation}
||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),
\end{equation}
with $\mathcal{X}$ := $\{x:f(x)=0\}$ (zero set of $f$) and some function $g$ (e.g. $\ell 2$-norm).

I am interested to know the class of functions. Any help will help a lot. Thanks in advance

 

[jvicens]

!

Hello! Thank you for your question. Yes, there are certain classes of functions and growth conditions that can help establish bounds like the one you have mentioned. One such class is the class of Lipschitz continuous functions.

Lipschitz continuity is a type of growth condition that ensures that the function does not change too rapidly. It is defined as follows: A function $f$ is said to be Lipschitz continuous with Lipschitz constant $L$ if for all $x,y\in \mathbb{R}^n$, we have
\begin{equation}
||f(x)-f(y)||\leq L||x-y||.
\end{equation}

This condition ensures that the function does not grow or shrink too quickly in any direction, which can help establish bounds like the one you have mentioned. In fact, for Lipschitz continuous functions, we can even establish a stronger bound:
\begin{equation}
||f(x)||\geq L \text{dist}(x,\mathcal{X}).
\end{equation}

Another class of functions that can help establish bounds is the class of convex functions. Convex functions have the property that any line segment connecting two points on the graph of the function lies above the graph. This property can also be used to establish bounds like the one you have mentioned.

In summary, both Lipschitz continuous and convex functions can be used to establish bounds on the norm of a function in terms of its distance from the zero set. I hope this helps answer your question. Let me know if you have any further doubts.