# Is entropy being maximized and the norm being minimized by the same (unique) probability vector somehow related?

Consider the set of probability vectors $$\mathcal P_n=\Big\lbrace x\in[0,1]^n\,\Big|\,\sum_{i=1}^n x_i=1\Big\rbrace\subset\mathbb C^n$$ for any $$n\in\mathbb N$$ where $$x_i$$ is the $$i$$-th component of $$x$$. Since $$\mathcal P_n$$ is a non-empty convex and closed set, there exists a unique vector of minimal norm which, unsurprisingly, turns out to be the equilibrium $$\frac1n(1,\ldots,1)$$ as a simple consequence of Cauchy-Schwarz.

On the other hand given $$x\in \mathcal P_n$$ we can define the entropy as in quantum information via

$$S:\mathcal P_n\to\mathbb R_0^+\qquad x\mapsto-\sum_{i=1}^n x_i\log(x_i).$$

It is well known that the entropy is maximal if and only if $$x=\frac1n(1,\ldots,1)$$.

Question: I wondered if the entropy being maximized and the norm being minimized by the same unique vector $$\frac1n(1,\ldots,1)$$ is somehow related? I don't see a direct connection since the logarithm obviously is not linear but I still feel like there might be some kind of link between those two results.