MEMO 2019 I2

Let $n\geq 3$ be an integer. We say that a vertex $A_i (1\leq i\leq n)$ of a convex polygon $A_1A_2 \dots A_n$ is Bohemian if its reflection with respect to the midpoint of $A_{i-1}A_{i+1}$ (with $A_0=A_n$ and $A_1=A_{n+1}$ ) lies inside or on the boundary of the polygon $A_1A_2\dots A_n$ . Determine the smallest possible number of Bohemian vertices a convex $n$ -gon can have (depending on $n$ ).

Proposed by Dominik Burek, Poland