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If $z_1,z_2,z_3,z_4 \in C$ satisfy $z_1+z_2+z_3+z_4=0$ and $|z_1|^2+|z_2|^2+|z_3|^2+|z_4|^2=1$, then what is the least value of $|z_1-z_2|^2+|z_2-z_3|^2+|z_3-z_4|^2+|z_4-z_1|^2$ ?

Here I can realize that the given set where we want to find the minimum value of the expression is the intersection of a hyper-sphere and a hyper-plane that is a sphere in short(check if I'm wrong?).By putting some random values like $z_1=1/2, z_2=-1/2, z_3=1/2, z_4=-1/2$ we can see that the value is coming 2. Can the value be further minimized? I here want to use minimum modulus principle and solve it but don't know from where and how do I start. This question is asked in TIFR-2012 Exam. Thank you for any help or suggestion.

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    It is not sure at all that the solution needs complex numbers. It can be re-casted as a $\mathbb{R^2}$ problem... – 2017-01-13
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    @JeanMarie if we convert problem into R^2 , then what would it make difference from complex problem. It will become more tedious to handle 8 variables – 2017-01-13

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