Let suppose you have a finite set of points $x_1, \ldots, x_n \in \mathbb{R}^d$.
The smallest circle problem is known as : \begin{equation} \underset{y \in \mathbb{R}^d}{\min}~\underset{i = 1, \ldots, n}{\max} \| x_i - y \|_2^2 \end{equation}
We know that the center of the smallest enclosing circle is in the convex hull of $x_1,\ldots,x_n$ (cf this question). We can rewrite this problem as :
\begin{equation} \underset{\substack{\alpha \in \mathbb{R}^n \\ \alpha \geq 0\\ \alpha^T \mathbf{1} = 1}}{\min}~\underset{i = 1, \ldots, n}{\max} \| x_i - \sum_{j=1}^n \alpha_j x_j \|_2^2 \end{equation}
I would like to consider a variation of this problem. Let $k$ be a fixed integer. I would like to know if there was an way of solving this problem if we add as an additional condition that $\forall i = 1, \ldots, n$, $\alpha_i$ must be of the form $\frac{\beta_i}{k}$ where $\beta_i \in \mathbb{N}$ :
\begin{equation} \underset{\substack{\beta \in \mathbb{N}^n \\ \beta^T \mathbf{1} = k}}{\min}~\underset{i = 1, \ldots, n}{\max} \| x_i - \frac{1}{k} \sum_{j=1}^n \beta_j x_j \|_2^2 \end{equation}