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This preference relation is convex. PROOF: suppose ''x'' and ''y'' are two equivalent bundles, i.e. <math>\min(x_1,x_2) = \min(y_1,y_2)</math>. If the minimum-quantity commodity in both bundles is the same (e.g. commodity 1), then this imples <math>x_1=y_1 \leq x_2,y_2</math>. Then, any weighted average also has the same amount of commodity 1, so any weighted average is equivalent to <math>x</math> and <math>y</math>. If the minimum commodity in each bundle is different (e.g. <math>x_1\leq x_2</math> but <math>y_1\geq y_2</math>), then this implies <math>x_1=y_2 \leq x_2,y_1</math>. Then <math>\theta x_1 + (1-\theta) y_1 \geq x_1</math> and <math>\theta x_2 + (1-\theta) y_2 \geq y_2</math>, so <math>\theta x + (1-\theta) y \succeq x,y</math>. This preference relation is convex, but not strictly-convex.
3. A preference relation represented by [[linear utility]] functions is convex, but not strictly convex. Whenever <math>x\sim y</math>, every convex combination of <math>x,y</math> is equivalent to any of them.
3. Consider a preference relation represented by:▼
:<math>u(x_1,x_2) = \max(x_1,x_2)</math>
This preference relation is not convex. PROOF: let <math>x=(3,5)</math> and <math>y=(5,3)</math>. Then <math>x\sim y</math> since both have utility 5. However, the convex combination <math>0.5 x + 0.5 y = (4,4)</math> is worse than both of them since its utility is 4.
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