The concurrent normals conjecture asserts that any convex body in n-dimensional Euclidean space has an interior point lying on normals through 2n distinct boundary points. It has been proved for \(n = 2,3\) (E. Heil) and, under a smoothness assumption, for \(n=4\) (J. Pardon). For \(n \geq 5\) , only the existence of an interior point on six normals is known. Zamfirescu has shown that most (in the Baire category sense) interior points of most convex bodies lie on infinitely many normals. We introduce a new approach by analyzing focal hypersurfaces of hedgehogs: we show for \(n = 3,4\) that any normal to a convex body K with smooth support function passes arbitrarily close to the interior points of \(K \cup L\) lying on normals through at least 6 boundary points of K, where L is the body bounded by the smallest convex parallel hypersurface to the boundary whose unit normal points in the opposite direction. Grebennilov and Pannina have extended this to \(n \geq 3\) via bifurcation theory. Finally, we prove for any convex body of \(\mathbb {R}^{4}\) it is false that there are at least 8 normal lines passing through the center of its minimal spherical shell.