Proper convex function Proper convex function In mathematical analysis (in particular convex analysis) and optimization, a proper convex function is a convex function "f" taking values in the extended real number line such that for at least one "x" and for every "x". That is, a convex function is "proper" if its effective domain is nonempty and it never attains formula_3. Convex functions that are not proper are called "improper convex functions". A "proper concave function" is any function "g" such that formula_4 is a proper convex function. For every proper convex function "f" on R there exist some "b" in R and
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