Suppose $a,b \in \mathbb{R}^n$ has nonnegative sorted entries, i.e., $$a_1 \geq a_2 \geq \ldots \geq a_n \geq 0~~\text{and}~~b_1 \geq b_2 \geq \ldots \geq b_n \geq b_{\min} >0.$$This indicates that $a$ may have 0 entries, while entries of $b$ are lower bounded by some constant $b_{\min}>0$. Further suppose $\|a\|_2 \geq 4\|b\|_2 >0$. What is a tight lower bound, i.e., the largest real $c > 0$ such that $$\|a \circ a - a \circ b \|_2 \geq c,$$where $\circ$ is Hadamard product, i.e., $a \circ b = [a_i b_i]_{i=1}^n$.
lower bound on the difference of Hadamard products
1
$\begingroup$
linear-algebra