Problem

Source: 2021ChinaTST test4 day2 P2

Tags: geometry, combinatorial geometry, combinatorics, area, Convex hull, convex polygon, symmetry



Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.