Suppose we are working in ${\mathbb R}^d$ (dimension is not fixed), and we have a set of $n$ points $X = \{x_1,\ldots,x_n\}$ in that space. Given a query point $y$ inside the convex hull of $X$, we want to choose $m < d + 1$ out of the $n$ points and take their convex hull and minimize the residual of a projection of $y$ onto the convex hull of the $m$ points, over all $n \choose m$ ways to choose $m$ points. If $d,n,m$ are part of the input, is this problem NP-hard?
EUCLIDEAN DISTANCE: The residual of a projection from $y$ to $x$ is the distance between two points $x$ and $y$ is $\sqrt{ \sum_{j=1}^d (x_j - y_j)^2}$