Prince Rupert's Problem: Geometric Foundations, Historical Background, and New Results on Convex Bodies

A cube can pass through a straight tunnel cut through another cube of the same size. From this seventeenth-century geometric paradox to recent counterexamples to the Rupert property, the problem has developed into a broader question concerning projections of convex bodies. The first part of this article reviews the historical background and introduces the essential geometric tools, including a vector-based reconstruction of the classical cube-passage configuration. The second part studies centred convex bodies whose support functions are expressed as spherical harmonic perturbations of the support function of a ball. For these bodies, we establish an exact passage criterion, construct explicit positive certificates for passage, and classify the three-parameter family\[T_{a,b,c}=aP_4(x)+bP_4(y)+cP_4(z) : \]the equality $D_2(T_{a,b,c}) = 0$ holds if and only if $ab = ac = bc = 0$. Proofs are included throughout.