konkavity

Konkavity is a term that appears in some mathematical and applied texts to describe a property related to concavity or non-convexity, but it is not standardized in mainstream references. In many instances, konkavity is simply an alternate or informal spelling of concavity. In other usages, authors employ konkavity as a broader or directional notion, requiring a form of concavity along specific directions or within certain subregions while relaxing the global condition.

Definition and interpretation vary. In the common interpretation, a real-valued function f defined on a convex domain is konkave if it satisfies the usual concavity inequality: for all x and y in the domain and for all t in [0,1], f(t x + (1−t) y) ≥ t f(x) + (1−t) f(y). Under this reading, konkave functions share standard properties of concave functions, such as the fact that local maxima are global maxima on convex domains. Some authors, however, use konkavity to describe a function that is concave along certain directions, or on particular subdomains, without holding globally.

Etymology and usage notes. The term derives from the same linguistic root as concave in several languages, and some writers retain the spelling konkavity as a stylistic or regional variant. Because its meaning is not universally fixed, readers should verify how a given text defines konkavity in its context.

See also: concavity, convexity, quasi-concavity, log-concavity.