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-3
1
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Let $v,w\in\mathbb{R}^{2}$ and $v\perp w$. Is it true that $\left\Vert v\right\Vert _{3}\leq\left\Vert v+w\right\Vert _{3}$, in which $\left\Vert \left(x,y\right)\right\Vert _{3}:=\sqrt[3]{\left|x\right|^{3}+\left|y\right|^{3}}$ for $\left(x,y\right)\in\mathbb{R}^{2}$? Thanks. |
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3
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So this asks for the general line L in the plane whether the minimum of the three-norm occurs at the point v obtained by dropping a perpendicular from 0 to L. Not true. Geometrically the tangents to the curves norm = constant rarely have their perpendiculars at the point of tangency passing through 0. |
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2
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No. Let $v=(x,y)$ suppose (without loss of generality) that both $x$ and $y$ are $>0$. Then $w=t(y,-x)$ for some $t$ and for small $t$ $$\|v+w\|_3^3= x^3 +y^3 +3t (x^2y-y^2 x) + 3t^2 (xy^2 +yx^2) + t^3 (y^3 -x^3).$$ So long as $x\neq y$ so that $x^2 y - y^2x \neq 0$, the inequality is violated for small $t$. |
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