Can we find a universal constant $c>0$ such that for all $p,q\in\Delta:=\lbrace x\in (0,1)^{n}\ \colon\ x\_{1}+\dots+x\_{n}=1\rbrace$ it is true that
\begin{equation}
|p\_{i}-q\_{i}|\le c\left|\ln\frac{p\_{i}}{g(p)}-\ln\frac{q\_{i}}{g(q)}\right|,\quad i=1,\dots,n
\end{equation}
where $g(p):=\left(\prod\_{i=1}^{n}p\_{i}\right)^{\frac{1}{n}}$ is the geometric mean of $p$? The question is of interest since it upper bounds the Euclidean distance on $\Delta$ with the Aitchison distance used in compositional data analysis.   

At least in dimension $n=2$ the answer is positive. Indeed, in that case the claim follows simply by the mean-value theorem:


\begin{align\*}
\frac{|p\_{1}-q\_{1}|}{|\ln\frac{p\_{1}}{g(p)}-\ln\frac{q\_{1}}{g(q)}|}=\frac{2|p\_{1}-q\_{1}|}{|\ln\frac{p\_{1}}{1-p\_{1}}-\ln\frac{q\_{1}}{1-q\_{1}}|}=2\xi(1-\xi)\le1,
\end{align\*}
where $\xi\in \overline{p\_{1}q\_{1}}$.   

Unfortunately, so far a failed to prove the general case (if true at all) and I'm grateful for any suggestions. Thanks in advance.



Lower bound for log-Ratios
Already in dimension $n=3$ the answer is negative.
Let $p=(1/3,1/3,1/3)$ and $q=(1/7,2/7,4/7)$. Then $g(p)=1/3$ and $g(q)=2/7$ so $p\_2/g(p)=q\_2/g(q)$.


