# inradius of convex surface with curvature upper bound

Let $M$ be a compact smooth convex surface bounding $V$ in $\mathbb{R}^3$. If the mean curvature $H$ (the arithmetic mean of principal curvatures) of $M$ is less that 1, can we put a ball of radius 1 inside $V$? How about if we assume the Gaussian curvature is less than 1?

• @FengWang Yes, 1/2 should be optional for $H$, but I do not have a proof. For Gauss curvature you can get arbitrary close to 0. – Anton Petrunin Mar 2 '17 at 16:08