Let $x_1, x_2, \ldots, x_n \in \mathbb{R}^p$ such that $\|x_i\|_2 \leq 1$. Does there exist $P: \mathbb{R}^q \to \mathbb{R}^p$, $q \geq p$, $\lambda > 0$ and $y_1, \ldots, y_n$ such that:
- $\|y_i\|_2 = 1$
- $P(y_i) = x_i$
- $\|y_i - y_j\|_2 = \lambda \cdot \|x_i - x_j\|_2$
- $P$ linear
The problem asks whether for any set of points within a unit hypersphere, there exists a set of points on the unit hypersphere in a higher dimensional space that projects onto the original points and preserves the scale of pairwise distances.