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A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [3] [4] [5] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph .
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
The function f(x, y) = x 2 − y 2 is concave-convex.. Von Neumann's minimax theorem can be generalized to domains that are compact and convex, and to functions that are concave in their first argument and convex in their second argument (known as concave-convex functions).
Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class. The convexity of a measure μ on n-dimensional Euclidean space R n in the sense above is closely related to the convexity of its probability density function. [2]
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
A conical combination is a linear combination with nonnegative coefficients. When a point is to be used as the reference origin for defining displacement vectors, then is a convex combination of points ,, …, if and only if the zero displacement is a non-trivial conical combination of their respective displacement vectors relative to .
This definition typically applies to trigonometric functions. [11] [12] The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620). [13] [14] concave function Is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.