We have a $n$-dimensional hypersurface $\Sigma$ embedded in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$. We know that $\Sigma$ is compact without boundary, convex (not necessarly strictly convex), and that the $k$-th symmetric polynomial in the principal curvatures is constant.

Can I deduce that $\Sigma$ is a sphere?

Note: in the case $k=1$ (maybe in the case $k=n$?) the answer is "yes".