A quadratic-density lower bound for tetrahedral cylinder covers

Let T be a regular tetrahedron and let A_min(T) be the least area of an orthogonal projection of T onto a plane. We prove that every finite cover of T by cylinders with measurable perpendicular bases B_i satisfies sum_i |B_i| >= (sqrt(15)/9) A_min(T). The proof computes the largest line integral of a radial quadratic density on T. We also prove that sqrt(15)/9 is the best constant obtainable from any nonnegative polynomial density of degree at most two by dividing its mass by a uniform bound on its line integrals. This optimality concerns the density method within that class, rather than the cylinder-covering infimum.