Is it true for 2 ≤ q ≤ p that ‖ x ‖ q ≤...

Keenan Santos

Keenan Santos



Is it true for 2 q p that
x q n p q p q x p
where x is an n-dimensional vector. I only need the inequality for n=2, so that would suffice. I'm just curious if it's true for any n. Just from plugging it into a graphing calculator I believe it to be true, but how would I prove it?

Answer & Explanation

Sophia Mcdowell

Sophia Mcdowell


2022-07-17Added 14 answers

x q p = ( 1 n i = 1 n | x i | q ) p / q n p / q
and using convexity of the map t t p / q , one gets
x q p 1 n i = 1 n | x i | p n p / q
or in other words,
x q p n p / q 1 x p p .

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