Logarithmically concave measure Logarithmically concave measure In mathematics, a Borel measure "μ" on "n"-dimensional Euclidean space formula_1 is called logarithmically concave (or log-concave for short) if, for any compact subsets "A" and "B" of formula_1 and 0 < "λ" < 1, one has where "λ" "A" + (1 − "λ") "B" denotes the Minkowski sum of "λ" "A" and (1 − "λ") "B". The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave. The restriction of the Lebesgue measure to any convex set is also log-concave. By a theorem of Borell, a measure is log-concave if and only if it has a density
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