Check if a point is in the interior of the convex hull of some other points in high dimensions, and lower-bounding the largest enclosed ball

Question

Given $m$ points $P=\{p_0, p_1, ..., p_m\}$ in high dimensions (e.g. 100), it is known that computing (or even representing) their convex hull $\text{conv}(P)$ is generally intractable due to the exponential dependence on dimensions. In this case, I wonder:

1. Is there an efficient algorithm to check whether some other given point $\tilde{p}$ is an interior point in the convex hull (i.e. there is a small ball centered at $\tilde{p}$ and lies in $\text{conv}(P)$)? I know we can construct a linear programming problem to check if a point lies inside the convex hull, but my question here is to further check if the convex hull has "volume" and if $\tilde{p}$ lies in its interior.
2. Following 1, can we compute or efficiently lower-bounding the largest enclosed (inscribed) ball centered at $\tilde{p}$ and lies in $\text{conv}(P)$?

I'm an amateur in geometry and I appreciate any potential solutions or suggestions. Thank you for your time!

The same question posted on mathoverflow.

Answer 1

Does the interior point algorithm finds an ellipsoid as the result? I mean we can define a convex program with being in convex hull of $P_i$s and objective function as the distance between the variable point and $\tilde{p}$. The interior point algorithm on this mathematical program will return an ellipsoid, if possible?