Radius of the largest ball with center $\boldsymbol{w}$, that can be inscribed in the Feasible region $\boldsymbol{F}$. Let $\displaystyle \boldsymbol{A}$ be an $\displaystyle m\times n$ matrix with  non zero rows, and let $\displaystyle \boldsymbol{b} \ \in \mathbb{R}^{m}$. Consider the feasible region $$\displaystyle \boldsymbol{F} \ =\{\boldsymbol{u} :\boldsymbol{Au} \geqslant \boldsymbol{b} ,\boldsymbol{u} \geqslant 0\}$$.Let $\displaystyle \boldsymbol{w}$ be an interior point of $\displaystyle \boldsymbol{F}$ Let $\displaystyle \delta (\boldsymbol{w})$ be the radius of the largest ball with center $\displaystyle \boldsymbol{w}$, that can be inscribed in $\displaystyle \boldsymbol{F}$. Find a formula for  $\displaystyle \delta (\boldsymbol{w})$.
My Try:
To understand the the question I am taking an example.
Let $\displaystyle \boldsymbol{A} \ =\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}$ and $\displaystyle \boldsymbol{b} \ =\ ( 0,0)$. Then the required feasible region is given by $$\displaystyle \boldsymbol{F} \ =\{\boldsymbol{u} :\boldsymbol{Au} \geqslant \boldsymbol{b} ,\boldsymbol{u} \geqslant 0\} \ $$  So
$$\begin{aligned}
\implies
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}\begin{bmatrix}
u_{1}\\
u_{2}
\end{bmatrix} \geqslant 0\\
u_{1} \geqslant \ 0\\
u_{2} \geqslant 0\ 
\end{aligned}$$
So, the feasible region is the First quadrant. Now we $\displaystyle \boldsymbol{w}$ is in first quadrant and it should be interior point.

How can I write the expression for the radius of the largest ball which is inscribed in the first quadrant i.e   $\displaystyle \delta (\boldsymbol{w})$.
As for understanding I took a particular example understand it.
How to solve the given question i.e, that is writing a  general expression for the   $\displaystyle \delta (\boldsymbol{w})$ by considering the   $\displaystyle \boldsymbol{A}$ be an $\displaystyle m\times n$ matrix with no non zero rows, and let $\displaystyle \boldsymbol{b} \ \in \mathbb{R}^{m}$

Consider a canonical representation of the set  $\boldsymbol{F}$ as an intersection of the family of halfspaces, that is $\boldsymbol{F}=\{\boldsymbol{u}:(\boldsymbol{u},\boldsymbol{a}_i)\ge b_i\mbox{ for each }i\}$, where all $\boldsymbol{a}_i $ are vectors of unit length and all $b_i$'s are non-negative real numbers. It is easy to check that $\displaystyle \delta (\boldsymbol{w})=\min_i |(\boldsymbol{u},\boldsymbol{a}_i)-b_i|$ 
is the smallest of distances from the point $\boldsymbol{w}$ to a hyperplane whose points satisfy an equation $(\boldsymbol{u},\boldsymbol{a}_i)=b_i$.
