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[수학 정보] 민코프스키 부등식
△π 2020.07.14 21:00 조회 1279
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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

민코프스키 부등식!

사실 이거 증명은 아래 '헬더의 부등식'만 알면 날로먹기입니다ㅋㅋㅋㅋㅋㅋ

민코프스키 부등식이란?

a_i >0\;\;b_i >0\;(i=1,2,3,\cdots,n),\;\;p>1일 때, 

\bigg(\sum_{i=1}^{n} (a_i +b_i)^{p}\bigg)^{\frac{1}{p}}\leq\bigg(\sum_{i=1}^{n}{a_i}^p\bigg)^{\frac{1}{p}}+\bigg(\sum_{i=1}^{n}{b_i}^p\bigg)^{\frac{1}{p}}이 성립한다!

 

자, 이걸 증명해봅시다!

먼저 헬더의 부등식을 다음과 같이 바꿔봅시다.

\sum_{i=1}^{n} x_i y_i\leq\bigg(\sum_{i=1}^{n}{x_i}^p\bigg)^{\frac{1}{p}}\bigg(\sum_{i=1}^{n}{y_i}^q\bigg)^{\frac{1}{q}}, \;\;\frac{1}{p}+\frac{1}{q}=1

여기서 x_i =a_i , \;\;y_i =(a_i +b_i )^{p-1}라 하면,

\sum_{i=1}^{n} a_i (a_i +b_i )^{p-1}\leq\bigg(\sum_{i=1}^{n}{a_i}^p\bigg)^{\frac{1}{p}}\bigg(\sum_{i=1}^{n}{(a_i +b_i )^{p}}\bigg)^{\frac{1}{q}}-----(1)이 되고, 마찬가지로 문자만 바꿔서

\sum_{i=1}^{n} b_i (a_i +b_i )^{p-1}\leq\bigg(\sum_{i=1}^{n}{b_i}^p\bigg)^{\frac{1}{p}}\bigg(\sum_{i=1}^{n}{(a_i +b_i )^{p}}\bigg)^{\frac{1}{q}}-----(2)식을 만들어 줍니다.

이제, 두 식을 더하면

\sum_{i=1}^{n} (a_i +b_i )^{p}\leq \left\{ \bigg(\sum_{i=1}^{n}{a_i}^p\bigg)^{\frac{1}{p}}+ \bigg(\sum_{i=1}^{n}{b_i}^p \bigg)^{\frac{1}{p}}\right\}\bigg(\sum_{i=1}^{n}{(a_i +b_i )^{p}}\bigg)^{\frac{1}{q}}

herefore\;\;\bigg(\sum_{i=1}^{n} (a_i +b_i)^{p}\bigg)^{\frac{1}{p}}\leq\bigg(\sum_{i=1}^{n}{a_i}^p\bigg)^{\frac{1}{p}}+\bigg(\sum_{i=1}^{n}{b_i}^p\bigg)^{\frac{1}{p}}\;\;(\because \frac{1}{p}=1-\frac{1}{q})

이로서 민코프스키 부등식이 증명되었습니다!!!

 

특히, p=2일 때, x=(a_1, a_2, c\cdots,a_n), y=(b_1 ,b_2 ,\cdots, b_n )인 벡터로 생각한다면,

||x+y||\leq||x||+||y||가 됩니다.

이렇게 민코프스키 부등식으로 삼각부등식을 증명할 수 있습니다!

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