Let $\emptyset \subsetneq S \subsetneq \{1,\cdots,n\}$ be a set with cardinality $s$, and $g\in\mathbb{R}^n$ be a vector such that 
$$\sum\_{\{i,j\}\subseteq \{1,\cdots,n\}}{(g\_i-g\_j)^2} = s(n-s).$$ 



> 
> **Question.** Is this true?
>  $$\sum\_{\{i,j\}\subseteq \partial S \\(g\_i-g\_j)^2\le1}{(g\_i-g\_j)^2} \ge \frac{s(n-s)}{n}$$ 
>  where $\partial S$ is the set of all $2$-subsets $\{i,j\}\subseteq \{1,\cdots,n\}$ that exactly one of $i$ or $j$ is in $S$. 
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Lower Bound for $\sum_{\{i,j\}\subseteq \partial S \\ (g_i-g_j)^2 \le1}{(g_i-g_j)^2}$
No. Take $n=3$, $S=\{3\}$, $g\_3=0$, $g\_2=a=1/1000$, $g\_1>1$ satisfies $f(g\_1)=(g\_1-a)^2+g\_1^2+a^2=2$, such $g\_1$ exists since $f(1)<2$, $f(2)>2$.


