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By definition, a convex combination of an indexed subset {v 0, v 1, . . . , v D} of a vector space is any weighted average λ 0 v 0 + λ 1 v 1 + . . . + λ D v D, for some indexed set of non‑negative real numbers {λ d} satisfying the equation λ 0 + λ 1 + . . . + λ D = 1. The definition of a convex set implies that the intersection of two ...
The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield. Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave ...
If a density is log-concave, so is its cumulative distribution function (CDF). If a multivariate density is log-concave, so is the marginal density over any subset of variables. The sum of two independent log-concave random variables is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.
Daniel Kahneman, who won the 2002 Nobel Memorial Prize in Economics for his work developing prospect theory. Prospect theory is a theory of behavioral economics, judgment and decision making that was developed by Daniel Kahneman and Amos Tversky in 1979. [1] The theory was cited in the decision to award Kahneman the 2002 Nobel Memorial Prize in ...
Right graph: With fixed probabilities of two alternative states 1 and 2, risk averse indifference curves over pairs of state-contingent outcomes are convex. In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter ...
A set of convex-shaped indifference curves displays convex preferences: Given a convex indifference curve containing the set of all bundles (of two or more goods) that are all viewed as equally desired, the set of all goods bundles that are viewed as being at least as desired as those on the indifference curve is a convex set.
Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces. Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
In economics, non-convexity refers to violations of the convexity assumptions of elementary economics.Basic economics textbooks concentrate on consumers with convex preferences (that do not prefer extremes to in-between values) and convex budget sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood.