In the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\), let \(K\) be a compact convex subset, and let \(K_1\) be the Steiner symmetral of \(K\) with respect to the hyperplane \(e_1^{\perp}\), orthogonal to a unit vector \(e_1\in\mathbb{R}^n\). Namely, \(K_1\) is obtained by translating all the chords of \(K\) in direction \(e_1\), so that their centers belong to \(e_1^{\perp}\). From a result by \textit{C. O. Kiselman} [J. Lond. Math. Soc., II. Ser. 33, 101--109 (1986; Zbl 0655.52004)], it follows that understanding the smoothness of \(K_1\) is a non trivial problem, since the Steiner symmetral of a convex body of class \(\mathcal{C}^{\infty}\) need not even be of class \(\mathcal{C}^2\). In this paper the main theorem shows that if \(K\) has \(\mathcal{C}^2\) boundary, and positive Gauss curvature, then the same holds for its Steiner symmetral. As a corollary, it follows that the orthogonal projection of \(K\) onto \(e_1^{\perp}\) also has, in the hyperplane \(e_1^{\perp}\), \(\mathcal{C}^2\) boundary and positive Gauss curvature. The question whether the theorem can be easily obtained from the corollary is left as an open problem, as well as the investigation of the same property for convex bodies having higher regularity class.