Inequality following from MAX and MIN

Question

Suppose I have the following

$x = \min(\max(y,z+w), k-w)$ and $z = \min(c, k-2w)$

where $x,y,z,w,k$ are real numbers and $w>0$.

Is it true that $z+w \le x$?

Answer 1

Let $u = z + w, v = k - w, r = c + w$.

Your question becomes

$u = \min(r,v), x = \min(\max(y,u),v)$. Is $u \le x$?

Here it is clear that either

• $x = \max(y, u)$ and therefore $x \ge u$, or
• $x = v$, and therefore $u = \min(r,v) = \min(r,x) \le x$.

In either case $u \le x$.

Therefore always $z + w \le x$