In economics, convex preferences are a property of utility functions commonly represented in an indifference curve as a bulge toward the origin for normal goods. (A concave utility function appears to bulge away from the origin instead.) It roughly corresponds to the "law" of diminishing marginal utility but uses modern theory to represent the concept.
Comparable to the greater-than-or-equal-to ordering relation \geq for real numbers, the notation \succeq below can be translated as: 'is as at least as good as' (in preference satisfaction). Formally, if \succeq is a preference relation on the consumption set X, then \succeq is convex if for any x, y, z \in X where y \succeq x and z \succeq x , then it is the case that \theta y + (1-\theta) z \succeq x for any \theta \in [0,1] .
\succeq is strictly convex if for any x, y, z \in X where y \succeq x and z \succeq x , and y \neq z then it is also true that \theta y + (1-\theta) z \succ x for any \theta \in (0,1) . It can be translated as: 'is better than relation' (in preference satisfaction).
An indifference curve displaying convex preferences thus means that the agent prefers, in terms of consumption bundles, averages over extremes.
References
Hal R. Varian. Intermediate Microeconomics A Modern Approach. New York: W.W. Norton & Company. ISBN 0-393-92702-4
Mas-Colell, Andreu; Whinston, Michael; & Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 978-0-19-507340-9
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