This is a comment. The result is clear for $n=1$ from the AM-GM inequality. The case $n=2$ can be shown by complex analysis: Represent the regular triangle by the roots of unity $1,\zeta,\zeta^2$, where $\zeta =-\frac 12+\frac {\sqrt{3}}2i$. Then the required function is represented by $$p(z)=|(z-1)(z-\zeta)(z-\zeta)|=|z^3-1|.$$ By the maximum modulus principle we may assume that $z$ lies on the side joining $\zeta$ to $\zeta^2$, i.e. $z$ is of the form $z=-\frac 12+ti,$ where $-\frac{\sqrt 3}2\leq t\leq\frac{\sqrt 3}2$. By calculus, the maximum happens when $t=0$ and $z=-1/2$, which is the midpoint of a side of the regular triangle.