Recently I am interested in Natural neighbor interpolation, that is :

Given a function $P(x)$ and some interpolation points $\{x_i,P(x_i)\}_{i=1}^N$, we have the interpolation function $$P^*(x)=\sum_{i=1}^N{\omega_i(x)P(x_i)},$$ where $\omega_i(x)=\dfrac{area\{\Omega_x\cap\Omega_{x_i}\}}{area\{\Omega_x\}},\Omega_x=\{y:\|y-x\|_2\leq\|y-x_k\|_2,k=1,\ldots,N\}.$

Now assume that $$L=\max_{j,k}\left\{\dfrac{|P(x_j)-P(x_k)|}{\|x_j-x_k\|}\right\},$$ my question is whether there exists a constant $C$ such that $$\sup_{X,Y}\left\{\dfrac{|P^*(X)-P^*(Y)|}{\|X-Y\|}\right\}\leq CL$$ holds, which means that the Lipschitz constant of the whole domain is bounded by that of the interpolation points?

Any counterexample or proof is welcome. Thank you.