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The triangle inequality is used in the derivations of many equations in physics. However, the triangle inequality is used in physics only it is basic form i.e. the sum of any two sides of a triangle is greater than the third side i.e. $a+b>c$. We do not actually care about the amount by which the sum of any two sides is greater than the third side. This is partly because not much is know the the mathematics literature about the quantity $a+b-c$ and partly because it may be be required to know value of $a+b-c$ from a physics point of view. Recently, I found three precise estimates for the triangle inequality proved as seen in this linked Mathematics Stack Exchange post and further improved to the form below:

Let $a \le b \le c$ be the sides of a triangle whose semi-perimeter is $s$ and area is $A$; then,

$$ a+b-c \ge \frac{8A^2}{s^3} \tag 1$$ $$ c+a-b \ge \frac{16A^2}{s^3} \tag 2$$ $$ b+c-a \ge \frac{18A^2}{s^3} \tag 3$$

with equality occurring only if the triangle is degenerate.

Given this precise estimates of the triangle inequality, I wanted to know if any equation in physics would benefit from it. For example there could be a left over or a delta term at the end of a physical equation which uses the triangle inequality somewhere which previous we were unbale to estimate accurately.

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    $\begingroup$ I mean, when we want a precise answer we typically can do trig and get an exact value, so this is only going to be useful if we only need an inequality, but with a tighter bound than the triangle inequality and trigonometric functions are too complicated...I can believe that could happen somewhere but it is going to be very niche $\endgroup$ May 18, 2023 at 8:52
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    $\begingroup$ Hard to say. Triangle inequality in complex numbers domain is used in quantum mechanics field, check this question or this article. $\endgroup$ May 18, 2023 at 10:54

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