This is a contribution to the topic of ‘mean curvature flow’ and its analogues, but for expanding instead of contracting convex hypersurfaces. The author studies the motion of a smooth, closed, uniformly convex hypersurface in Euclidean space \(\mathbb{R}^{n+1}\) expanding in the direction of the outer unit normal vector, with speed given by a suitable function of the principal radii of curvature. The latter function is assumed to be symmetric, positive, homogeneous of degree one, and concave. It is proved that a smooth solution exists, the hypersurface remains smooth and uniformly convex for all time and that after suitable normalization it converges to a round sphere. The author’s approach involves studying the evolution equation satisfied by the support function of the expanding hypersurface.