Given $k$ arbitrary points $P_1, P_2, ..., P_k$ on the unit sphere in $\mathbb{R}^n$. Consider the function $f(P)=|PP_1|.|PP_2|...|PP_k|$, where $P \in \mathbb{R}^n$. Is it true that the maximum of $f$ on the unit sphere is larger than 1?
For the circle, we can think of $f(P)$ as the modulus of a holomorphic function on $\mathbb{C}$, and the conclusion follows easily. However, this argument cannot be applied to higher dimensions.
For more details ($n=2$), let $z_{i}$ be arbitrary points with $|z_{i}|=1$. Consider $p(z)=(z-z_{1})\cdots (z-z_{k})$. We have $|p(0)|=|z_{1}\cdots z_{k}|=1$, so by maximum modulus principle, we get $\max _{|z|=1} |p(z)|\geq |p(0)|=1$.