Convex function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Real-valued_function/jetqlJpK">real-valued function</a> is called convex if the <a href="/facts/Line_segment/V8E0M4qk">line segment</a> between any two distinct points on the <a href="/facts/Graph_of_a_function/htYX0BGX">graph of the function</a> lies above or on the graph between the two points. Equivalently, a function is convex if its <a href="/facts/Epigraph_(mathematics)/qGYcHa8t">epigraph</a> (the set of points on or above the graph of the function) is a <a href="/facts/Convex_set/vdAuJRJl">convex set</a>. 
In simple terms, a convex function graph is shaped like a cup 
 
 
 
 ∪
 
 
 {\displaystyle \cup }
 
 (or a straight line like a linear function), while a <a href="/facts/Concave_function/HxVEBkce">concave function</a>'s graph is shaped like a cap 
 
 
 
 ∩
 
 
 {\displaystyle \cap }
 
.
A twice-<a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> function of a single variable is convex <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> its <a href="/facts/Second_derivative/3aA6j4EY">second derivative</a> is nonnegative on its entire <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>. Well-known examples of convex functions of a single variable include a <a href="/facts/Linear_function/Ejvy8CKy">linear function</a> 
 
 
 
 f
 (
 x
 )
 =
 c
 x
 
 
 {\displaystyle f(x)=cx}
 
 (where 
 
 
 
 c
 
 
 {\displaystyle c}
 
 is a <a href="/facts/Real_number/R02gw5Pb">real number</a>), a <a href="/facts/Quadratic_function/mKxHIC3U">quadratic function</a> 
 
 
 
 c
 
 x
 
 2
 
 
 
 
 {\displaystyle cx^{2}}
 
 (
 
 
 
 c
 
 
 {\displaystyle c}
 
 as a nonnegative real number) and an <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> 
 
 
 
 c
 
 e
 
 x
 
 
 
 
 {\displaystyle ce^{x}}
 
 (
 
 
 
 c
 
 
 {\displaystyle c}
 
 as a nonnegative real number).
Convex functions play an important role in many areas of mathematics. They are especially important in the study of <a href="/facts/Optimization/oRn8Iv5I">optimization</a> problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an <a href="/facts/Open_set/QfTCIqMu">open set</a> has no more than one <a href="/facts/Maximum_and_minimum/ADlgnyzV">minimum</a>. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a>. In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, a convex function applied to the <a href="/facts/Expected_value/1XV0JKL8">expected value</a> of a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> is always bounded above by the expected value of the convex function of the random variable. This result, known as <a href="/facts/Jensen%2527s_inequality/sosRQWN1">Jensen's inequality</a>, can be used to deduce <a href="/facts/Inequality_(mathematics)/qkze4zdI">inequalities</a> such as the <a href="/facts/Inequality_of_arithmetic_and_geometric_means/STAHxc1l">arithmetic–geometric mean inequality</a> and <a href="/facts/H%25C3%25B6lder%2527s_inequality/7I1ubTWw">Hölder's inequality</a>.

Convex function open-in-new

Convex function