Does convexification preserve Lipschitz continuity?

Question

I am facing the following problem:

Consider a $d$-Lipschitz continuous and convex function $f$ defined on $\Delta$ that is the simplex of dimension $n \in \mathbb{N}^*$. Let $G$ be a finite grid of pairs point-value : $$ G = \{(b_1, f(b_1)),...,(b_k, f(b_k)) \} $$ where $b_i \in\Delta$ for all $i \in [1, k]$.

Now consider $\bar{f}$ the convexification of $f$ restricted to the pairs of the grid $G$. $\bar{f}$ is defined on $\Delta$ as follows: $$ \bar{f}(b) = \min_{\lambda} \sum_{i =1}^k\lambda_if(b_i) $$ where $\sum_i \lambda_i = 1$ and $\sum_i \lambda_i b_i = b$

I would like to prove that the function $\bar{f}$ is $d$-Lipschitz continuous on the simplex (for the 1-norm: $||x||_1 = \sum_i|x_i|$)

Does anyone recommend any litterature or have any hints?

Regards

Answer 1

Here is an easy counterexample: Consider $f(x,y) = y^2$ and restrict $f$ to the points $$ (0,0), (1-\varepsilon, 0), (1,1), (1,-1) $$ Then, the Lipschitz constant of the convexification is $\varepsilon^{-1}$.