Cauchy-Schwarz tells you that $$\sum_{i=1}^n |A_iA_{i+1}|^2\geq \frac{P^2}{n}$$ where $P$ is the perimeter of the polygon. Then we need the inequality $$P^2\geq 4n\tan(\pi/n)A$$ which is the classical isoperimetric inequality for polygons with many proofs in the literature, analytic, geometric and algebraic. See this article and its references, for example.

Cauchy–Schwarz tells you that $$\sum_{i=1}^n \lvert A_iA_{i+1}\rvert^2\geq \frac{P^2}{n}$$ where $P$ is the perimeter of the polygon. Then we need the inequality $$P^2\geq 4n\tan(\pi/n)A$$ which is the classical isoperimetric inequality for polygons with many proofs in the literature, analytic, geometric and algebraic. See Fan, Taussky, and Todd - An algebraic proof of the isoperimetric inequality for polygons article and its references, for example.

Cauchy Schwartz tells you that $$\sum_{i=1}^n |A_iA_{i+1}|^2\geq \frac{P^2}{n}$$ where $P$ is the perimeter of the polygon. Then we need the inequality $$P^2\geq 4n\tan(\pi/n)A$$ which is the classical isoperimetric inequality for polygons with many proofs in the literature, analytic, geometric and algebraic. See this article and its references, for example.

Cauchy-Schwarz tells you that $$\sum_{i=1}^n |A_iA_{i+1}|^2\geq \frac{P^2}{n}$$ where $P$ is the perimeter of the polygon. Then we need the inequality $$P^2\geq 4n\tan(\pi/n)A$$ which is the classical isoperimetric inequality for polygons with many proofs in the literature, analytic, geometric and algebraic. See this article and its references, for example.