Concave

Concave describes a surface or shape that curves inward. In two dimensions, a shape is concave if it is not convex; more precisely, there exist two points in the shape such that the straight line segment joining them intersects the exterior of the shape. For polygons, concavity is often diagnosed by interior angles: a polygon is concave if at least one interior angle exceeds 180 degrees.

In mathematics, concavity also applies to functions. A function is concave on an interval if its graph lies on or above every chord between two points of the graph: f(tx+(1-t)y) ≥ t f(x) + (1-t) f(y) for t in [0,1]. If a function is twice differentiable, concavity is indicated by f''(x) ≤ 0 on that interval.

In optics, a concave surface curves inward. A concave lens (diverging lens) is thinner at the center and makes light rays diverge. A concave mirror reflects off an inward-curving surface and can form real or virtual images depending on the object position.

In computational geometry and related fields, the concave hull is a boundary intended to fit a set of points more tightly than the convex hull. It is not uniquely defined and depends on the chosen algorithm, often requiring parameter choices to balance tightness against noise or holes.