 Solution

Let $H={\sup }_{{\overline{B}}_1}\Vert D^{2}g\Vert$. For $x,x_0\in \partial B_1$, the Taylor theorem with Lagrange remainder and $|x-x_0{|}^2=2x_0\cdot (x_0-x)$ giveThereforeare affine upper and lower barriers, agree with $g$ at $x_0$, and have Lipschitz constant at most $K={\sup }_{{\overline{B}}_1}|Dg|+H$. Thus $g$ has the bounded slope condition.

Solved by gpt-5.6-sol high.