10.3. Find all points on the plane, with coordinates $(x, y)$, that satisfy the equation: $\max \{|x+y|,|x-y|\}=1$
Answer. Rhombus with vertices $((-1,0),(1,0),(0,-1),(0,1))$.

Solution. Let's expand the absolute values of the expressions. In the diagram, the upper one indicates $|x+y|$, the lower one - the modulus $|x-y|$, and the maximum of the expressions is framed:

![](https://cdn.mathpix.com/cropped/2024_05_06_2ab06a7e3e15d67c48a3g-2.jpg?height=514&width=585&top_left_y=617&top_left_x=724)