DSE 2020 Question 49: Concavity, convexity and quasi-concavity

Given nonempty subsets $Y_1,\dots,Y_n$ of $\mathbb{R}^2$, let $Y^*=\left\{\sum_{j=1}^{n}y_j\mid y_1\in Y_1,\dots,y_n\in Y_n\right\}$. Fix $p\in\mathbb{R}^2$, and for a nonempty $X\subseteq\mathbb{R}^2$ let $v(p,X)=\sup\{p\cdot x\mid x\in X\}$. Suppose there is $y^*\in Y^*$ with $p\cdot y^*=v(p,Y^*)$, and for every $i$ there is $y_i\in Y_i$ with $p\cdot y_i=v(p,Y_i)$. Then