# Why is the negative entropy Lipschitz with respect to the $1$-norm (Over)?

Let $\left\|x \right\| = \sum_{i=1}^{i=n}\left|x^i\right|$ and $d\left(x\right)=\sum_{i=1}^{i=n}x^i\ln x^i$ where $x\in R^n$ and $\sum_{i=1}^{i=n}x^i=1$

How to prove: For all $x, x'$, $$\left| d\left(x\right)-d(x') \right|\leq \frac{1}{2}\left\|x-x' \right\|$$

By the way, the above question is equivalent to the question below. Let $\left\|x \right\| = \sum_{i=1}^{i=n}\left|x^i\right|$ and $d\left(x\right)=\sum_{i=1}^{i=n}x^i\ln x^i$, where $x\in \left\{R^n \mid x\ge0,\sum_{i=1}^{i=n}x^i=1\right\}$ The question is how to prove $d_1\left(x\right)\ge \frac{1}{2}\left\|x-x_0\right\|$ where $x_0= \operatorname{argmin}_x \left\{d_1\left(x\right)\right\}$. Actually $x_0^i=\frac{1}{2},i=1,2,...,n$