One can minimize $\sum_j \lambda_j^2$ subject to the constraints listed in the question. Then the minimizer has all entries non-zero if and only if $p$ is in the interior.
This is convex quadratic programming, can be done in polynomial time, solvers readily are available, if you actually need to solve these kinds of problems.

Even better is to use an interior point method to solve the linear programming problem of minimizing $\sum_i a_i\lambda_i$ for a generic $a$, subject to the constraints listed. Then it's not so hard to see (it's a property of these interior point methods) that the optimal λ will be positive iff $p$ is interior.