Given (i) αx + βy, α + β = 1, α, β ≥ 0 (ii) αx + βy, α + β > 1 (iii) (1 – α)x + (1 – β)y, 0 < α < 1 0 <…

Both (i) and (iv) are examples of a convex combination of $x$ and $y$. A convex combination of $x$ and $y$ is a linear combination with non-negative weights whose sum is $1$: in (i), $\alpha, \beta \geq 0$ and $\alpha + \beta = 1$, and in (iv), $0 \leq \alpha \leq 1$ ensures $\alpha \geq 0$, $1 - \alpha \geq 0$, and $\alpha + (1 - \alpha)$ $= 1$. Expression (ii) is not a convex combination because the weights sum to more than $1$, and (iii) is not a convex combination in general because $(1 - \alpha) + (1 - \beta)$ $= 2 - (\alpha + \beta)$ need not equal $1$.