Exercise 13.13 - Lower bounds to convex functions

Since $f$ is a convex function, then any hyperplane that is tangent to an arbitrary point ${\mathbf{\text{𝐱}}}^{\prime }\in \text{dom}(f)$ has the entire $f$ above it. Denote the normal of the hyperplane at ${\mathbf{\text{𝐱}}}^{\prime }$ by ${\mathbf{\text{𝐧}}}_{\mathbf{\text{𝐱}}}^{\prime }$, then the hyperplane intersect with $\mathbf{\text{𝐱}}$ at $(\mathbf{\text{𝐱}},{\varphi }_{\mathbf{\text{𝐱}}}^{\prime })$ with:

And we have:

This assertation holds for arbitrary ${\mathbf{\text{𝐱}}}^{\prime }$, hence:

For more details, refer to Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation.