Given $m$ vectors $x_{i| i=1\cdots m}$ and $y_{i| i=1\cdots m}$. Is there any inequality relationship between the max of the product of the square of $\ell_2$ norm of $m$ vectors and the product of the max of the square of $\ell_2$ norm of $m$ vectors i.e
Is there a relationship exist
$$ \max_{i} x_ix'_i \max_j y_jy'_j \leq \max_i x_ix'_iy_iy'_i $$ here $x'_i$ is the transpose of vector $x_i$.
if so, can anyone give me a proof
Let $a_i=x_ix_i’$ and $b_i=y_iy_i’$. Then $a_i,b_i$ are non-negative reals and we may show $$\max a_i \max b_i \geqslant \max a_ib_i$$
One way to see this is to plot $a_i$ and $b_i$ on the Cartesian plane. The LHS is the largest rectangle (by area) while RHS represents the area of a rectangle contained within it. Equality is possible when the same $i$ maximises $a_i$ and $b_i$ simultaneously.