Inequality involving the min function

Question

I'm trying to prove the following inequality: $$ \left|y_{1}\land x_{1}-y_{2}\land x_{2}\right|\leq\left|y_{1}-y_{2}\right|+\left|x_{1}-x_{2}\right|, $$ where $x\land y=\mbox{min}(x,y)$. By considering different cases it seems to hold but I haven't been able to find a simple way of proving it. I'd appreciate any suggestions.

Answer 1

Hint: Consider the four cases:

• $y_1<x_1$ and $y_2<x_2$

• $y_1<x_1$ and $y_2\geq x_2$

• $y_1\geq x_1$ and $y_2<x_2$

• $y_1\geq x_1$ and $y_2\geq x_2$

By symmetry, you only need to consider the first two cases.

• The first case is simple because the LHS is a term on the RHS.

• For the second case, either $y_1<x_2<y_2$ so that $|y_1-y_2|=|y_1-x_2|+|x_2-y_2|$ or $x_2<y_1<x_1$ so $|x_2-x_1|=|x_2-y_1|+|y_1-x_1|$. In either case, we get that $|y_1-x_2|$ is less than or equal to a term on the RHS.

Answer 2

(Using the approach from Is $|f(a) - f(b)| \leqslant |g(a) - g(b)| + |h(a) - h(b)|$? when $f = \max\{{g, h}\}$):

If $x_2 \le y_2$ then $$ y_{1}\land x_{1}-y_{2}\land x_{2} \le x_1 - x_2 \le \lvert x_1 - x_2\rvert \le \max \{ \lvert x_1 - x_2 \rvert, \lvert y_1 - y_2 \rvert \} $$ and the same is true if $x_2 \ge y_2$. By symmetry it follows that $$ \lvert y_{1}\land x_{1}-y_{2}\land x_{2} \rvert \le \max \{ \lvert y_1 - y_2 \rvert, \lvert x_1 - x_2 \rvert \} $$ which is slightly stronger than the desired $$ \lvert y_{1}\land x_{1}-y_{2}\land x_{2} \rvert \le \lvert y_1 - y_2 \rvert + \lvert x_1 - x_2 \rvert \quad . $$

Answer 3

Here is an exotic, one-line-r based on $$\min\left ( x,y \right )=\frac{1}{2}\left ( x+y - \left | x-y \right | \right )$$

$$\left | \min\left ( x_1,y_1 \right ) - \min\left ( x_2,y_2 \right ) \right |=\left | \frac{1}{2}\left ( x_1+y_1-\left | x_1-y_1 \right | \right ) - \frac{1}{2}\left ( x_2+y_2-\left | x_2-y_2 \right | \right ) \right |=$$

$$=\left | \frac{1}{2}\left ( x_1-x_2 \right ) + \frac{1}{2}\left ( y_1-y_2 \right ) + \frac{1}{2}(\left | x_2-y_2 \right | - \left | x_1-y_1 \right |) \right | \leq$$ $$\leq \frac{1}{2}\left | x_1-x_2 \right | + \frac{1}{2}\left | y_1-y_2 \right | + \frac{1}{2}\left | \left | x_2-y_2 \right | - \left | x_1-y_1 \right | \right |\leq $$

$$\leq \frac{1}{2}\left | x_1-x_2 \right | + \frac{1}{2}\left | y_1-y_2 \right | + \frac{1}{2}\left | \left ( x_2-y_2 \right ) - \left ( x_1-y_1 \right ) \right |= $$

$$= \frac{1}{2}\left | x_1-x_2 \right | + \frac{1}{2}\left | y_1-y_2 \right | + \frac{1}{2}\left | \left ( x_2-x_1 \right ) + \left ( y_1-y_2 \right ) \right |\leq$$

$$\leq \frac{1}{2}\left | x_1-x_2 \right | + \frac{1}{2}\left | y_1-y_2 \right | + \frac{1}{2} \left | x_2-x_1 \right | + \frac{1}{2} \left | y_1-y_2 \right | = \left | x_1-x_2 \right | + \left | y_1-y_2 \right | $$