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.)
The accepted answer thoroughly answers the question, but here is an informal way to think about it.
In the inequality $x^2 + y^2 \geq Kxy$ that you gave, you can think of the right-hand side as a bound or barrier for the left-hand side. Tightness means the barrier actually touches the thing it's containing, so you cannot move the barrier any closer to the thing it's containing.
There is no gap. It is tight.
