konkávní

Konkávní is a Czech term meaning concave. It is used in mathematics, geometry, optics, and everyday language to describe inward-curving forms. The word derives from Latin concavus and in Czech appears as konkávní with diacritical marks.

In geometry, konkávní describes shapes that are not convex. A konkávní polygon has at least one interior angle greater than 180 degrees, or equivalently, a line segment joining two points inside the polygon may pass outside the polygon. Such shapes are contrasted with convex polygons, where every line segment between two points in the polygon lies entirely inside it.

In mathematics, konkávní is used for concave functions. A function f defined on a convex domain is konkávní if for all x and y in the domain and all t in [0, 1], f(t x + (1 − t) y) ≥ t f(x) + (1 − t) f(y). Equivalently, for any two points on the graph, the line segment joining them lies below the graph. For twice differentiable functions, concavity corresponds to f'' ≤ 0. Concave functions play a central role in optimization, economics, and risk assessment, with Jensen’s inequality expressing that for a concave f, E[f(X)] ≤ f(E[X]) for a random variable X.

In optics, konkávní zrcadlo refers to a concave mirror with an inward-curved reflecting surface that can converge parallel rays to a focal point. Its behavior depends on curvature and focal length; such mirrors are used in telescopes and various optical devices.

See also: konvexní.