Regularised 1 norm and distance
The 1-norm of a vector and the distance between two vectors and in are given respectively, see Figure fig:XO:norms by

Left: The unitspheres of the three classic norms. Right: comparison with the regularised 1 norm of parameter .
Their regularised counterpart is made such that it is possible to
differentiate the expression and avoid divisions by zero by introducing
a parameter $\epsilon$. The corresponding definitions are
```{math}
||\va||_{1,\epsilon}
\DEF\sum_{i=1}^n \frac{a_i^2}{\sqrt{a_i^2+\epsilon^2}},
\qquad
d_{1,\epsilon}(\va,\vb)\DEF\sum_{i=1}^n \frac{(a_i-b_i)^2}{\sqrt{(a_i-b_i)^2+\epsilon^2}}.