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The probability density function of the normal distribution is quasiconcave but not concave. The bivariate normal joint density is quasiconcave. In mathematics , a quasiconvex function is a real -valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form ( − ∞ , a ...
Already in 1952, Morrey conjectured that rank-1-convexity does not imply quasiconvexity. [10] This was a major unsolved problem in the Calculus of Variations, until Šverák gave an counterexample in 1993 for the case d ≥ 2 {\displaystyle d\geq 2} and m ≥ 3 {\displaystyle m\geq 3} . [ 11 ]
The following are among the properties of log-concave distributions: If a density is log-concave, so is its cumulative distribution function (CDF). If a multivariate density is log-concave, so is the marginal density over any subset of variables. The sum of two independent log-concave random variables is log-concave. This follows from the fact ...
In mathematics, concavification is the process of converting a non-concave function to a concave function. A related concept is convexification – converting a non-convex function to a convex function. It is especially important in economics and mathematical optimization. [1]
Points where concavity changes (between concave and convex) are inflection points. [5] If f is twice-differentiable, then f is concave if and only if f ′′ is non-positive (or, informally, if the "acceleration" is non-positive). If f ′′ is negative then f is strictly concave, but the converse is not true, as shown by f(x) = −x 4.
The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms. Given a set X, a convexity over X is a collection 𝒞 of subsets of X satisfying the following axioms: [9] [10] [23] The empty set and X are in 𝒞; The intersection of any collection from 𝒞 is in 𝒞.
The concept of strong convexity extends and parametrizes the notion of strict convexity. Intuitively, a strongly-convex function is a function that grows as fast as a quadratic function. [11] A strongly convex function is also strictly convex, but not vice versa.
Convex and Concave - a print by Escher in which many of the structure's features can be seen as both convex shapes and concave impressions. Convex body - a compact convex set in a Euclidean space whose interior is non-empty. Convex conjugate - a dual of a real functional in a vector space. Can be interpreted as an encoding of the convex hull of ...