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 $M$.
Problem
Source: 2021ChinaTST test4 day2 P2
Tags: geometry, combinatorial geometry, combinatorics, area, Convex hull, convex polygon, symmetry
25.04.2021 08:44
The answer is easy , 2! Prove: Find a rectangle covers P with area not bigger than 2, and chose M be the centre of that rectangle. Then Q is also covered by it, so the convex hull of P and Q is covered by rectangle,whose area not bigger than 2. Construction: A isosceles right triangle, need some discussion.
25.04.2021 09:46
Actually, you proved $\alpha\le 2.$ But $2$ is not the minimum value. I believe $\alpha=\frac{4}{3}$ and this value is attained when $P$ is a triangle. The optimal point $M$ is the centroid of $P.$
25.04.2021 10:59
Note that the problem is about "the convex hull of $P \cup Q$ ", not $P \cup Q$.
25.04.2021 15:49
Sorry, my bad, I've read it as "the area of $P\cup Q$" not "the area of the convex hull of $P \cup Q$ ". But, could someone check the translation again. I belive it should be "the area of $P\cup Q$". Because, otherwise the problem is next to trivial, and I do not believe a problem like that to be given at a Chinese TST.
26.04.2021 04:49
dgrozev wrote: Sorry, my bad, I've read it as "the area of $P\cup Q$" not "the area of the convex hull of $P \cup Q$ ". But, could someone check the translation again. I belive it should be "the area of $P\cup Q$". Because, otherwise the problem is next to trivial, and I do not believe a problem like that to be given at a Chinese TST. In fact, it's "the convex hull of $P \cup Q$".Besides, many problems in Chinese TST are trivial.
07.01.2022 15:53
I think the problem is not at all trivial. The idea of considering such a rectangle appears quite ingenious to me (though, guessing that the answer is $2$ seems not tough because of case of a triangle).