Varian's Intermediate Microeconomics describes convexity as $$\text{Given } x, y \in X: x \sim y \implies \forall t \in [0,1], tx + (1-t)y \succeq x,y$$
The other definition I read everywhere is: $$\text{Given } x, y \in X: x \succeq y \implies \forall t \in [0,1], tx + (1-t) y \succeq y$$
Consider $X = \mathbb{R}^{2}$. The first definition does not imply the second when we have, for example, $U(x_1,x_2) = \begin{cases} 0 \text{ if } x_1 + x_2 = 1 \\ 2 \text{ if } x_1 + x_2 > 1 \\ 1 \text{ if } x_1 + x_2 < 1 \end{cases}$.
Assume $X = \mathbb{R}^2$ and preferences are complete, transitive and strictly monotone. (Strict monotonicity is defined as $y \geq x \ (y \neq x) \implies y \succ x$ where $(y_1, y_2) = y \geq x = (x_1, x_2)$ means $y_1 \geq x \land y_2 \geq x_2$.) Following are two questions that I thought of and was unable to prove/disprove :
- Can we have a preference relation that satisfies the three assumptions and the first definition of convexity but not the second one?
- Can we have a utility function that describes a preference relation which satisfies the three assumptions and the first definition of convexity but not the second one?