Suppose $P\in \mathbb{R}^{n\times n}$ is a positive definite symmetrical matrix. $F\in \mathbb{R}^{m\times n}$, $g\in \mathbb{R}^{m}$, $n > m$. The set $S=\{x|Fx=g\}$ is an $n-m$ dimensional hyperplane.
Is there a name for the following stuff?
$$min\{\sqrt{x^TPx}|x\in S\}$$
It would be the Euler distance from $S$ to the origin point if $P=I^{n\times n}$.
It is the Mahalanobis distance. Formally, given a test point $x$ and a probability distribution $\mathcal{D}$ with mean $\mu$ and covariance $C$, the Mahalanobis distance between $x$ and $\mathcal{D}$ is measured as $$ \text{dist}_M(x, \mathcal{D}) = \sqrt{(x-\mu)^TC^{-1}(x-\mu)} $$ Because $P$ in the question is positive definite, its inverse exists and $P^{-1}$ serves as $C$ above.