In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to...

In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it...

like a linear function), while a concave function's graph is shaped like a cap ∩ {\displaystyle \cap } . A twice-differentiable function of a single variable...

that is neither convex nor continuous. Convex function Concave function Logarithmically concave function Pseudoconvexity in the sense of several complex...

-\infty .} Convex functions that are not proper are called improper convex functions. A proper concave function is by definition, any function g : X → [ − ∞...

Schur-convex functions are used in the study of majorization. A function f is 'Schur-concave' if its negative, −f, is Schur-convex. Every function that is...

Look up concave or concavity in Wiktionary, the free dictionary. Concave or concavity may refer to: Concave lens Concave mirror Concave function, the negative...

of sublinear growth: every function f ( n ) ∈ o ( n ) {\displaystyle f(n)\in o(n)} can be upper-bounded by a concave function of sublinear growth. Asymmetric...

this usage. The complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context...

studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex...

function: f ( x , y ) = − ( cos ⁡ ( x 2 ) + cos ⁡ ( y 2 ) ) 2 . {\displaystyle f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}.} Asymptote Chart Plot Concave function...

optimizing a convex or concave function can also be described as the problem of maximizing or minimizing a submodular function subject to some constraints...

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...

characterize Euler's gamma function among the possible extensions of the factorial function to real arguments. Logarithmically concave function Kingman, J.F.C. 1961...

interval of Z. These conditions mirror the ones required for log-concave functions. Sequences that fulfill the three conditions are also called Pólya...

an interior corner is a line of concave function, whereas a fillet on an exterior corner is a line of convex function (in these cases, fillets are typically...

monotone function is an important type of real-valued function, fully classified by Charles Löwner in 1934. It is closely allied to the operator concave and...

case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa...

relationship between the expenditure function and the utility function. If given a specific regular quasi-concave utility function, the corresponding price is...

} is a continuous function that is concave-convex, i.e. f ( ⋅ , y ) : X → R {\displaystyle f(\cdot ,y):X\to \mathbb {R} } is concave for every fixed y...

been called the "concave" Hollow Earth hypothesis or skycentrism. Cyrus Teed, a doctor from upstate New York, proposed such a concave Hollow Earth in 1869...

Log-concave may refer to: Logarithmically concave function Logarithmically concave measure Logarithmically concave sequence This disambiguation page lists...

lens with two concave surfaces is biconcave (or just concave). If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the...

Because the values are in decreasing order, the cumulative function is a concave function. To take the example below, in order to lower the amount of...

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle:...

linear function is a convex function, which implies that every local minimum is a global minimum; similarly, a linear function is a concave function, which...

downward bias, by Jensen's inequality, due to the square root's being a concave function. The bias in the variance is easily corrected, but the bias from the...

theory of convex functions named after Werner Fenchel. Let ƒ be a proper convex function on Rn and let g be a proper concave function on Rn. Then, if regularity...

ISBN 978-3-540-42669-1. Tangian, Andranik (2002). "Constructing a quasi-concave quadratic objective function from interviewing a decision maker". European Journal of...

function. Thus, any Gaussian measure is log-concave. The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave....