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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.

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    $\begingroup$ Is it not the case that there are multiple solutions to $p_i/g(p) = const$ when n > 2 ? A dimension counting argument would suggest this is the case. Then $|p_i - q_i| > 0 $ but the rhs = 0 $\endgroup$ – mike Oct 17 '19 at 12:42
  • $\begingroup$ I'm not sure. Taking logs in $p_{i}=cg(p)$ leads to the linear system $(\operatorname{id}-\frac{1}{n}\mathbf{1}\otimes\mathbf{1})x=\ln c\mathbf{1}$, which as far as I see admits solutions only if $c=1$. In that case solutions are of the form $a\mathbf{1}$ for some $a$. Then transforming back ,we end up with the apparently only solution $p=\frac{1}{n}\mathbf{1}$ since we require $p\in\Delta$. $\endgroup$ – Tobsn Oct 17 '19 at 13:56
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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 $p_2/g(p)=q_2/g(q)$.

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