The $\|\cdot\|\_{\infty}$-norm on $\mathbb{R}^n$ for $n\in \mathbb{Z}^+$ is not a smooth function. However, I came across [this post](https://stats.stackexchange.com/questions/298849/soft-version-of-the-maximum-function) which essentially says that a pointwise approximation to the maximum function, and therefore $\|\cdot\|\_{\infty}$ is
$$
m\_{\lambda}(x) = \sum\_{i=1}^n\,\frac{e^{\lambda\, x\_i}x\_i}{\sum\_{j=1}^n\,e^{\lambda\, x\_j}}
$$
where $x=(x\_i)\_{i=1}^n\in \mathbb{R}^n$ is arbitrary.


Fix $\lambda>0$ and consider the map $d\_{\lambda}:\mathbb{R}^n\times \mathbb{R}^n\rightarrow [0,\infty)$ defined by
$$
d\_{\lambda}(x,y) = m\_{\lambda}(|x-y|)
$$
where $|z|:=(|z\_i|)\_{i=1}^n$ for any $z\in \mathbb{R}^n$.


Does $d\_{\lambda}$ define a quasi-metric on $\mathbb{R}^n$?


It certainly satisfies positivity, symmetry and $d\_{\lambda}(x,y)=0$ if and only if $x=y$ by its not obvious that it should satisfy a relaxed triangle inequality: ie
$$
d\_{\lambda}(x,y) \le C\big(d\_{\lambda}(x,z)+d\_{\lambda}(z,y)\big)
$$
for every $x,y,z\in \mathbb{R}^n$ and some $C\ge 1$ independent of $x,y,z$.



Is this a smooth approximation to the $\ell$-infinity distance actually a quasi-metric?
$\newcommand\R{\mathbb R}$By rescaling, without loss of generality $\lambda=1$.


So, the question becomes the following: is there a real $C$ not depending on $u=(u\_1,\dots,u\_n)\in\R\_+^n$, $v=(v\_1,\dots,v\_n)\in\R\_+^n$, and $w=(w\_1,\dots,w\_n)\in\R\_+^n$ such that for all such $u,v,w$
$$w\le u+v\implies m(w)\le C(m(u)+m(v)) \tag{1}\label{1},$$
where $w\le u+v$ means $w\_i\le u\_i+v\_i$ for all $i\in[n]:=\{1,\dots,n\}$ and $m(u):=\dfrac{\sum\_{i\in[n]}e^{u\_i}u\_i}{\sum\_{i\in[n]}e^{u\_i}}$?


Clearly, $m(u)\le\max u:=\max\_{i\in[n]}u\_i$ for $u\in\R\_+^n$. Also, by (say) [Chebyshev's sum inequality](https://en.wikipedia.org/wiki/Chebyshev%27s_sum_inequality), $m(u)\ge\bar u:=\frac1n\sum\_{i\in[n]}u\_i$ for $u\in\R\_+^n$. So, for all $u,v,w$ in $\R\_+^n$ such that $w\le u+v$,
$$m(w)\le\max w\le\max u+\max v\le n\bar u+n\bar v\le n m(u)+n m(v),$$
so that \eqref{1} holds with $C=n$.


