Area covered by convex polygon centered at vertices of the unit square

by Xin Yuan Li   Last Updated January 16, 2018 14:20 PM

Let $U$ be a convex symmetric polygon. Put a copy of $U$ at each vertex of the unit square. Let $t$ be the smallest positive real number such that $\mathcal{U} = \cup_{\mathbf{a}\in \{0,1\}^2} tU + \mathbf{a}$ covers the unit square. See figure below.

Now if I let $\mathcal{U}' = \cup_{\mathbf{a}\in \{0,1\}^2} \frac{t}{2}U + \mathbf{a}$ (as shown in the rightmost picture) is it true that $\mathcal{U}'$ covers $\leq \frac{1}{2}$ of the area of the unit square?

Convexity is crucial here since there are counter-examples for non-convex polygons. I would still like to try to solve this problem, so I will accept answers which contain relevant theorems or resources (papers, textbooks, lecture notes).

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