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⪰ 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?

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    $\begingroup$ 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. $\endgroup$ Commented Dec 11, 2023 at 14:47
  • $\begingroup$ So that ⪰ is a striclty convex preference relation on X = (R_+)^N does not make sense? $\endgroup$ Commented Dec 13, 2023 at 17:24
  • $\begingroup$ It does, but not on the set of natural numbers. $\endgroup$ Commented Dec 13, 2023 at 17:25
  • $\begingroup$ does (R_+)^N mean all positive real numbers or all natural numbers? $\endgroup$ Commented Dec 13, 2023 at 17:27
  • $\begingroup$ Neither. It is the set of all N-tuples of nonnegative real numbers. $\endgroup$ Commented Dec 13, 2023 at 17:38

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