# Proof for convex preference relation

⪰ is a strictly convex preference relation on a set X which is the set of all N-tuples of nonnegative real numbers. Further are x,y,z element of this set X and i have the preferences x ≻ y ≻ z given.

How can i show that there exists a unique t ∈ R with y = tx + (1 − t)z?

My approach is the following. I have a convex linear combination for y which lies between x and z, which is also represented in the preferences. I guess this is true for every t ∈ (0,1). So in between this open interval there is a unique solution. But how can i show that in detail?

• Strictly convex preferences only make sense on a convex set. This rules out the natural numbers. And the result will generally not hold on convex domains either. The lexicographic ordering on $\mathbb{R}^2$ is strictly convex but violates your condition. Dec 11, 2023 at 14:47
• So that ⪰ is a striclty convex preference relation on X = (R_+)^N does not make sense? Dec 13, 2023 at 17:24
• It does, but not on the set of natural numbers. Dec 13, 2023 at 17:25
• does (R_+)^N mean all positive real numbers or all natural numbers? Dec 13, 2023 at 17:27
• Neither. It is the set of all N-tuples of nonnegative real numbers. Dec 13, 2023 at 17:38