Let $e=(1,\ldots,1)^T$, the unit simplex be $\triangle = \{x\in\mathbb{R}^n: \sum_{i=1}^nx_i=1,x\geq0\},$ with centroid $c = e/n$, and $B_r(c) = \{ x\in\mathbb{R}^n:\|x-c\| \leq r\}$. Then the problem is to find the largest radius $r$ of the inscribed ball $\triangle\cap B_r(c)$.
My question is not about how to solve the problem, but rather how to formulate it as a solvable optimization problem? I've found many answers which provide solutions: Radius of inscribed sphere of n-simplex., radius of the inscribed sphere of the pyramid in $\mathbb{R^5}$, Constrained Problem, Largest inscribed ball, but I've yet to find any information on how to actually formulate it as an optimization problem of the form \begin{align} \text{minimize}\quad& f(x)\\ \text{subject to}\quad& h_i(x) = 0,i=1,\ldots,m,\\ &g_j(x) \leq 0,j=1,\ldots,l. \end{align}
On the surface, I think the problem can be stated as \begin{align} \text{maximize}\quad& r\\ \text{subject to}\quad& r\leq\|x-c\|,\\ & e^Tx =1,\\ & x\geq 0, \end{align} however this is does not seem correct to me since $e_1=(1,0,\ldots,0)^T$ is a feasible solution and $\|e_1 - e/n\|=\sqrt{(1-1/n)^2+\frac{n-1}{n^2}}=\sqrt{\frac{n-1}{n}} >\frac{1}{\sqrt{n(n-1)}}.$ It is obvious that I miss some constraints that include the faces of the simplex, but I'm not sure how to formulate them. Any help in this regard would be appreciated!
If I understood the problem correctly, we want to maximize the radius of the inscribed $n-1$ dimensional ball formed by the intersection of a $n-$ ball of arbitrary size about the $n-$ unit simplex's centroid and the simplex, such that the intersection is a $n-1$ - ball. We can deduce that if every point on the shell of the $n-1$ - ball lies on the intersection, then the entire ball must also, and that the $n-1$ - ball must have the same radius as the $n-$ ball.
So we can formulate as, \begin{align} \max_{R} \quad &R \\ \text{s.t.} \quad &e^Tx = 1 \, \text{and} \, x\geq 0 \quad \forall\, x = c+Rv \; \text{s.t.}||v||_2 = 1 \\ &R \geq 0 \end{align}
Simplifying, \begin{align} \max_{R} \quad &R \\ \text{s.t.} \quad &e^T(c+Rv) = 1 \quad &&\forall\, ||v||_2 = 1 \\ &c+Rv \geq 0 \quad &&\forall\, ||v||_2 = 1 \\ &R \geq 0 \end{align}
I don't know how to properly implement the $\forall\,||v||_2 = 1$ part in an optimization formulation. One idea is to numerically discrete the space of $||v||_2 = 1$, and check the constraint only for the discretization. Another idea is embedded optimization \begin{align} \max_{R} \quad &R \\ \text{s.t.} \quad &\left[\min_{v : ||v||_2 = 1 } e^T(c+Rv)\right] = 1 \\ &\left[\max_{v : ||v||_2 = 1 } e^T(c+Rv)\right] = 1 \\ &\left[\min_{v : ||v||_2 = 1 } c+Rv\right] \geq 0 \\ &R \geq 0 \end{align}