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For example, a real-valued Lebesgue measurable function that is midpoint-convex is convex: this is a theorem of Sierpiński. [8] In particular, a continuous function that is midpoint convex will be convex.
is a convex set. [2] The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis. Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures.
Convex conjugate - a dual of a real functional in a vector space. Can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes. Convex curve - a plane curve that lies entirely on one side of each of its supporting lines. The interior of a closed convex curve is a convex set.
The symmetry of is the reason and are identical in this example. In mathematics (in particular, functional analysis ), convolution is a mathematical operation on two functions ( f {\displaystyle f} and g {\displaystyle g} ) that produces a third function ( f ∗ g {\displaystyle f*g} ).
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, [1] whereas mathematical optimization is in general NP-hard. [2 ...
An arrangement in which the midpoint of all the spheres lie on a single straight line is called a sausage packing, as the convex hull has a sausage-like shape.An approximate example in real life is the packing of tennis balls in a tube, though the ends must be rounded for the tube to coincide with the actual convex hull.
In geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space , or equivalently as the set of all convex combinations of points in the subset.
For every proper convex function : [,], there exist some and such that ()for every .. The sum of two proper convex functions is convex, but not necessarily proper. [4] For instance if the sets and are non-empty convex sets in the vector space, then the characteristic functions and are proper convex functions, but if = then + is identically equal to +.