A preference relation $\succ_i$ is strictly convex if for any two bundles $x$ and $y$ in the choice set $X$, and for any $\alpha \in (0,1)$, if $x \succ_i y$, then $\alpha x + (1-\alpha)y \succ_i y$.
Now, let's assume that the preference relation $\succ_i$ is
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