Let ≿ be a convex preference relation on a linear space X. The set of best elements
X* ={x : x ≿ y for every y ∈ X} is convex.
A slightly stronger notion of convexity is often convenient (example 1.116). A preference relation is strictly convex if averages are strictly preferred to extremes. Formally the preference relation ≿ is strictly convex if for every x; y ∈ X with x≿y but x≠y,
ax + (1 - a) for every 0 < a < 1