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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. Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave ...
An example of a concave polygon. A simple polygon that is not convex is called concave, [1] non-convex [2] or reentrant. [3] A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. [4]
As an example, consider the function () = on the domain .This function is quasiconcave, but it is not concave (in fact, it is strictly convex). It can be concavified, for example, using the monotone transformation () = /, since (()) = is concave.
Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x 2 /2 is a concave function of x. But f is not concave since the second derivative is positive for | x | > 1:
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 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 ...
A concave mirror, or converging mirror, has a reflecting surface that is recessed inward (away from the incident light). Concave mirrors reflect light inward to one focal point. They are used to focus light. Unlike convex mirrors, concave mirrors show different image types depending on the distance between the object and the mirror.
Examples of real images include the image produced on a detector in the rear of a camera, and the image produced on an eyeball retina (the camera and eye focus light through an internal convex lens). In ray diagrams (such as the images on the right), real rays of light are always represented by full, solid lines; perceived or extrapolated rays ...