Let $\mathbf{B}$ be a definite positive square matrix of size $n \times n$, and $\mathbf{b}$ an $n$-sized vector. It can be shown that the solution of $\arg\min_x \left(\mathbf{x}^T \mathbf{B} \mathbf{x} - 2\mathbf{b}^T \mathbf{x}\right)$ is $\mathbf{B}^{-1} \mathbf{b}$. Consequently, we can assert that $\mathbf{b} \mapsto \arg\min_x \left(\mathbf{x}^T \mathbf{B} \mathbf{x} - 2\mathbf{b}^T \mathbf{x}\right)$ constitutes a linear mapping from $\mathbb{R}^n$ to $\mathbb{R}^n$.
However, when we introduce a positivity constraint, i.e., when we seek the solution in the set of non-negative vectors, I suspect the application $\mathbf{b} \mapsto \arg\min_{\mathbf{x} \geq 0} \left(\mathbf{x}^T \mathbf{B} \mathbf{x} - 2\mathbf{b}^T \mathbf{x}\right)$ to be a piecewise linear function, as my numerical experiments show. What approaches or methods can be pursued to demonstrate this type of result?
EDIT:
1D case, $B$ is a strictly positive number. Then $b \mapsto \arg\min_{\mathbf{x} \geq 0} \left(B x^2-2bx\right)$ is simply $\max(0, b/B)$
