I defined an Euclidean norm function with a smooth technique to avoid not differentiable problem in point $0$, $f: \mathbb{R}^n\to\mathbb{R}$, $$f(x) = \sqrt{\|x\|^2+\epsilon^2},$$ where $\epsilon=0.00001$.
I know for sure that $f(x)$ is convex, so $\nabla^2 f(x) \ \forall x\in \mathbb{R}^n$ positive semi-definite.
My question: is the Hessian matrix $\nabla^2 f(x) \ \forall x\in \mathbb{R}^n$ positive definite?
I scaled down the problem to 1-dimension, such that we can write down the gradient(first derivative) and the Hessian( derivative) easily, $$f'(x) = \frac{x}{\sqrt{(x^2+\epsilon^2)}}$$ $$f''(x) = \frac{\epsilon^2}{(x^2+\epsilon^2)\sqrt{(x^2+\epsilon^2)}}.$$
It follows $f''(x)>0 \ \forall x\in\mathbb{R}$, which make it a positive definite matrix.
Can we generalize it to high dimensions $n$?
$ \def\e{\varepsilon} \def\l{\left} \def\r{\right} \def\p{\partial} \def\lr#1{\l(#1\r)} \def\trace#1{\operatorname{Tr}\lr{#1}} \def\grad#1#2{\frac{\p #1}{\p #2}} \def\hess#1#2#3{\frac{\p^2 #1}{\p #2\,\p #3^T}} $Let $x$ be a vector in ${\mathbb R}^{n}$.
Square the function before calculating the differential and gradient of $f$ $$\eqalign{ f^2 &= x^Tx + \e^2 \\ 2f\,df &= 2x^Tdx \\ df &= \lr{\frac{x}{f}}^Tdx \\ \grad{f}{x} &= \frac xf \;=\; g &\quad\quad\big({\rm Gradient\,vector}\big) \\\\ }$$ Now calculate the differential and gradient of $g$ $$\eqalign{ dg &= \frac{f\,dx-x\,df}{f^2} \\ &= \frac{fI\,dx-x\lr{\frac xf}^T\,dx}{f^2} \\ &= \lr{\frac{f^2I-xx^T}{f^3} }dx \\ \grad{g}{x} &= \frac{f^2I-xx^T}{f^3} \;=\; H &\qquad\big({\rm Hessian\,matrix}\big) \\ }$$ Note that for vectors orthogonal to $x$, the action of the Hessian matrix is simple linear scaling $$Hv = {\frac{v}{f}} \qquad\qquad\qquad\qquad\qquad $$
But vectors which are almost parallel to $x$ are scaled more aggressively.
In the limiting case $$Hx = {\frac{(f^2-x^Tx)x}{f^3}} \;=\; {\frac{\e^2x}{f^3}} \qquad$$
Setting $\,\e^2=0\;$ would make $Hx=0,\;$ and the matrix would be semi-definite.
However, since $\e^2>0\;$ the matrix is positive-definite.