Convex measure same thing as it being a logarithmically concave measure. The classes of "s"-convex measures form a nested increasing family as "s" decreases to −∞" or, equivalently Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class. The convexity of a measure "μ" on "n"-dimensional Euclidean space R in the sense above is closely related to the convexity of its probability density function. Indeed, "μ" is "s"-convex if and only if there is an absolutely continuous measure "ν" with probability density function "ρ" on some R so that
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