 Solution

The eigenvalues $p_j$ of a density operator are nonnegative and sum to one. ThereforeFor a non-pure state at least two eigenvalues are positive, so the last expression is strictly positive. Combining this with the Generalized Bloch representation givesThus every non-pure quantum state lies strictly inside the radius sphere for $N\ge 2$. It need not be an interior point of the entire physical state body: for $N>2$, a rank-deficient mixed density operator can lie on that body's boundary while still lying inside the Euclidean sphere.