Isoperimetric quotients for a decomposed convex body (Q5932810)

Let \(K\subset{\mathbb R}^2\) denote a convex body which is decomposed into convex bodies \(K_1,\ldots,K_n\) (\(n\geq 2\)) with areas \(A_1,\ldots,A_n\) and perimeters \(L_1,\ldots,L_n\). Then the authors prove the sharp isoperimetric inequality \((L_1+\cdots +L_n) /(\sqrt{A_1}+\cdots+\sqrt{A_n})> 2 \cdot 12^{1/4}\) if \(\min A_i/\max A_i\) is bounded from below by a suitable constant. The lower bound for this isoperimetric ratio is equal to the perimeter of a regular hexagon of unit area. This result extends a previous inequality due to G. and L. Fejes Tóth which was established for convex polygons \(K\) with at most 6 sides. In a similar vein, the authors also show that \(L_1^2+\cdots +L_n^2>4\sqrt{12} (A_1+\cdots+A_n)\).