Convex preferences quasi-concave utility functions, although these are not necessary for the analysis of preferences. Convex preferences In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly corresponds to the concept of diminishing marginal utility without requiring utility functions. Comparable to the greater-than-or-equal-to ordering relation formula_1 for real numbers, the notation formula_2 below can be translated as: 'is at least as good as' (in preference satisfaction). Similarly, formula_3 can be translated as 'is strictly better than'
Luneburg lens one side, and formula_14 only on the other: The parameter formula_19 is a constant for any given ray, but differs between rays passing at different distances from the centre of the lens. For rays passing through the centre, it is zero. In some special cases, such as for Maxwell's fish-eye, this first order equation can be further integrated to give a formula for formula_14 as a function or formula_13. In general it provides the relative rates of change of formula_14 and formula_13, which may be integrated numerically to follow the path of the ray through the lens. Luneburg lens A