You have a point $P(x_1,...,x_n)$ and an arbitrary function $f(x_1,...,x_n) = c$ . Is finding the minimum distance from the point to the function/curve possible in realistic time?
I mean something like this, but in any dimension: Shortest distance from point to curve
Minimizing the distance function in 1000 dimension is not a possibility I think.
The general from of a curve, surface, etc. in a finite-dimensional space is given by a map $f : U \to \Bbb B^n$, where $U \subseteq \Bbb R^m$ is a region (connected non-empty open set). The "hypersurface" is the image of $U$ under $f$, that is, the set $f(U)$.
Assuming $f$ is differentiable, only a small variation of the technique in the other thread is needed. If $p$ is the point whose distance from the hypersurface is desired, one forms the squared distance function $$D(x) = \|f(x) - p\|^2$$ where $x = (x_1, x_2, ..., x_m)$. $D$ is differentiable because $f$ is, and at the minimum, the derivative with respect to each of the coordinates must be $0$. So you solve the system of equations
$$\frac {\partial D}{\partial x_1} = 0\\\frac {\partial D}{\partial x_2} = 0\\\vdots\\\frac {\partial D}{\partial x_m} = 0$$ This gives you a system of $m$ equations in the $m$ unknowns $x_1, ..., x_m$, which you then solve to minimize $D$.
While this might be impractical to do by hand for a 1000-dimensional problem, it is not impossible.
