TITLE: Lower bounds for inner product $x^\top y$
QUESTION [7 upvotes]: Cauchy-Schwarz provides an upper bound for the inner product $x^\top y$. Are there theorems that talk about lower bounds for this quantity? Assume $x\ge 0$ and $y\ge 0$ wlog.

REPLY [3 votes]: There are some theorems regarding lower bound on inner product. One of them is due to Kantorovich, $\Vert x \Vert \Vert y \Vert \le (\frac{A}{G}) x^T y$ for $x_i,y_i\gt 0$. Where $0\lt m\le \frac{x_i}{y_i} \le M \lt \infty$,  $A=(m+M)/2$ and $G=\sqrt (mM)$.