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Minkowski's inequality says the following: For every sequence of scalars $a = (a_i)$ and $b = (b_i)$, and for $1 \leq p \leq \infty$ we have: $||a+b||_{p} \leq ||a||_{p}+ ||b||_{p}$. Note that $||x||_{p} = \left(\smash{\sum\limits_{i=1}^{\infty}} |x_i|^{p}\right)^{1/p}$. This is how I tried proving it: \begin{align*} ||a+b||^{p} &= \sum |a_k+b_k|^{p}\\\ &\leq \sum(|a_k|+|b_k|)^{p}\\\ &= \sum(|a_k|+|b_k|)^{p-1}|a_k|+ \sum(|a_k|+|b_k|)^{p-1}|b_k|. \end{align*}

From here, how would you proceed? I know that you need to use Hölder's inequality. So maybe we can bound both the sums on the RHS since they are products.

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instead of using $\displaymath...$ you can use $$...$$ to get displayed math; and macros like \begin{align} to get aligned equations. –  Arturo Magidin Jan 5 '11 at 2:02

2 Answers 2

up vote 4 down vote accepted

Holder's Inequality would say that $$\sum |x_ky_k| \leq \left(\sum |x_k|^r\right)^{1/r}\left(\sum|y_k|^s\right)^{1/s}$$ where $\frac{1}{r}+\frac{1}{s}=1$.

Apply Holder's twice, once to each sum, using $x_k = a_k$, $y_k = (|a_k|+|b_k|)^{p-1}$ in one, and similarly in the other, with $r=p$ and $\frac{1}{s}=1-\frac{1}{p}$.

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There's a version of the proof using exponential's convexity. I let you try to find it by yourself and will complete my post later to give the solution.

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