Question 7
Do there exist four points $P_{i} = (x_{i}, y_{i}) \in \mathbb{R}^{2}$ ($1 \leq i \leq 4$) on the plane such that:
- for all $i = 1,2,3,4$, the inequality $x_{i}^{4} + y_{i}^{4} \leq x_{i}^{3} + y_{i}^{3}$ holds, and
- for all $i \neq j$, the distance between $P_{i}$ and $P_{j}$ is greater than $1$?
- for all $i = 1,2,3,4$, the inequality $x_{i}^{4} + y_{i}^{4} \leq x_{i}^{3} + y_{i}^{3}$ holds, and
- for all $i \neq j$, the distance between $P_{i}$ and $P_{j}$ is greater than $1$?