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
Lens (optics) object off the optical axis of the lens is imaged, where rays pass through the lens at an angle to the axis θ. Rays that pass through the centre of a lens of focal length "f" are focused at a point with distance from the axis. Rays passing through the outer margins of the lens are focused at different points, either further from the axis (positive coma) or closer to the axis (negative coma). In general, a bundle of parallel rays passing through the lens at a fixed distance from the centre of the lens are focused to a ring-shaped