= Solution
The <eigenvalues> $p_j$ of a <density operator> are nonnegative and sum to one. Therefore
$$
1-\operatorname{Tr}(\rho^2)=\left(\sum_jp_j\right)^2-\sum_jp_j^2=2\sum_{j<k}p_jp_k.
$$
For a non-pure state at least two <eigenvalues> are positive, so the last expression is strictly positive. Combining this with the <Generalized Bloch representation> gives
$$
\boxed{\|s\|^2=\operatorname{Tr}(\rho^2)-\frac1N<1-\frac1N.}
$$
Thus every non-pure <quantum state> lies strictly inside the radius sphere for $N\geq2$. 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.
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