# Extension of the Triangle Inequality

• Caspian
In summary, the conversation discusses the triangle inequality and its extension, which states that |a + b|^p \le 2^p (|a|^p + |b|^p). The speaker is unsure how to prove or disprove this inequality and suggests using the binomial theorem. Another speaker mentions that it follows from the convexity of xp for p > 1.
Caspian
Pretty much knows the triangle inequality.
$$\left| a + b \right| \le \left| a \right| + \left| b \right|$$

I was reading a source which asserted the following extension of the triangle inequality:
$$\left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right)$$

This is bugging me because I can't figure out how to prove or disprove it. It's sensible enough that it might actually be true... but want to know for sure whether it's true.

I tried proving it by induction, but it got really messy...

Is this inequality true? Or, is it wrong? Thanks!

when you tried induction, did you try the binomial theorem?

It follows from convexity of xp for p > 1.
$$(\frac{1}{2}|a|+\frac{1}{2}|b|)^p\leq \frac{1}{2}|a|^p+\frac{1}{2}|b|^p$$
$$(|a|+|b|)^p\leq 2^{p-1}(|a|^p+|b|^p)$$

## 1. What is the Triangle Inequality?

The Triangle Inequality is a mathematical principle that states the sum of any two sides of a triangle must be greater than the third side.

## 2. What is an Extension of the Triangle Inequality?

An extension of the Triangle Inequality is a variation of the original principle that applies to more complex geometric shapes, such as polygons or higher-dimensional shapes.

## 3. How does the Extension of the Triangle Inequality differ from the original?

The Extension of the Triangle Inequality takes into account additional factors, such as the number of sides or dimensions, and may involve more complex equations or proofs.

## 4. What are some real-world applications of the Extension of the Triangle Inequality?

The Extension of the Triangle Inequality is used in fields such as physics, engineering, and computer science to determine the feasibility of constructing certain structures or to ensure the accuracy of measurements and calculations.

## 5. Why is the Extension of the Triangle Inequality important in mathematics?

The Extension of the Triangle Inequality is a fundamental principle in mathematics that helps to establish the validity and accuracy of geometric proofs and calculations. It also has practical applications in various fields, making it a valuable tool for problem-solving and analysis.

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