I want to calculate whether a point $P$ lies a convex hull or not in high dimensions, e.g. $n = 50$. I am aware of the linear programming approach that boils down to solve the following equation for each point $P$ and the convex hull spanning points $P_{i}$:
$P = \sum_{i=1}^{n} \lambda _{i}P_{i}$
Unfortunately, this approach does not scale well for high dimensions. On my search for a more efficient solution I stumbled across the following thread: How to know if a point in a $n$ dimensional space is inside an area? Since the guidelines forbid to ask questions there I am forced to create a new thread.
The approach the user sebigu proposed is not very clear to me although I bought the recommended book and read the corresponding paragraphs. He proposed to look at the supporting hyperplanes and the inner normals. What benefit does it hold to inspect these? Let's assume the connection between two supporting points of the convex hull span a supporting hyperplane and I can construct a normal between $P$ and the supporting hyperplane. What value does this add? How can I conclude from there whether $P$ lies inside the convex hull or not?
Thank you in advance
Kind regards