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) real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image, "S" then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a magnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the lens of the eye to create a "real image" on the retina.