# Proving the uniform convexity of $L^p$ for $1 < p \le 2$

We are asked to show the uniform convexity of $$L^p$$ for $$1 < p \le 2$$ using the following inequality:

For all $$1 < p < \infty$$, there is a constant $$C$$ such that $$|a - b|^p \leq C(|a|^p + |b|^p)^{1-s}\big(|a|^p + |b|^p - 2|\frac{a + b}{2}|^p\big)^s$$ where $$s = p/2$$ for all $$a, b \in \mathbb{R}$$

I have actually managed to prove this inequality and I think reaching uniform convexity from this step should be easy but I'm having trouble with this final step!

I think we're supposed to use this inequality above with Holder's inequality but I'm unsure how exactly.

• Related Apr 27, 2022 at 22:02