= Solution
Let $H=\sup_{\overline B_1}\lVert D^2g\rVert$. For $x,x_0\in\partial B_1$, the <Taylor theorem with Lagrange remainder> and $|x-x_0|^2=2x_0\mathbin\cdot(x_0-x)$ give
$$
|g(x)-g(x_0)-Dg(x_0)\mathbin\cdot(x-x_0)|
\leq Hx_0\mathbin\cdot(x_0-x).
$$
Therefore
$$
b_{x_0}^{\pm}(x)=g(x_0)+Dg(x_0)\mathbin\cdot(x-x_0)
\pm Hx_0\mathbin\cdot(x_0-x)
$$
are 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>.
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