Meaning of tightness of inequality I'd like to better understand the idea of tightness of an inequality. I found this helpful post but would like to know more. For example, is tightness only changed by modifying coefficients in a linear equation? For continuity, here's the example used in the post mentioned: > $$ (x,y>0) \\\ x^2+y^2\geq Kxy $$ Is "tight" for K=2. If K>2 then it fails. If K<2 then it can be improved to chose K=2.

An inequality is _tight_ if there is some choice of the variables involved for which **equality** holds. Otherwise it is not. For instance, for positive $x, y$ the inequality

$$x^2 + y^2 \geq 2 x y$$

is true, and moreover if you take $x = y = 1$ then both sides equal 2, so this inequality is tight.

The inequality

$$x^2 + y^2 \geq x y$$

is also true for positive $x, y$, but it is _not_ tight, since we always have (for instance)

$$x^2 + y^2 \geq 2 x y > x y$$

There can be many ways to make an inequality which isn't tight into one which is. For instance the (true) inequality

$$x + 1 \geq x$$

is not tight. We could "modify" it to the tight inequality

$$x \geq x.$$

(There's no formal notion of modifying an inequality to another one; it's just meant in the loose, intuitive sense.)