Lower bound on Sum of squared differences

Question

Consider $n+2$ real numbers $x_i$ with $0 \leq x_i \leq \frac{1}{2}$. Additionally, not all $x_i$ are the same.

Now define two quantities

$\Phi = 4\sum_{i=0}^{n+1}x_i^2$

$\Phi' = \sum_{i=1}^{n}(x_{i-1} - x_{i+1})^2 + (x_0 - x_1)^2 + (x_{n} - x_{n+1})^2$.

I need a lower bound on $\Phi'$.

Is it true that there is a constant c, such that $\Phi' > c \cdot \Phi$?

How does $c$ look like?

Answer 1

I think that you cannot obtain such a bound. Namely, if you choose $x_k = a + k \epsilon$, with $a\in (0,1/2)$ and $\epsilon>0$ very small (compared to $a$), you get $$ \Phi \simeq 4(n+2)a^2, \qquad \Phi' = (4n-6)\epsilon^2. $$