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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c0262401.png" />, defined on some interval, satisfying the condition
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c0262402.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A function $f$, defined on some interval, satisfying the condition
  
for every two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c0262403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c0262404.png" /> from this interval. The geometrical meaning of this condition is that the midpoint of any chord of the graph of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c0262405.png" /> is located either above the graph or on it. If the inequality (1) is strict for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c0262406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c0262407.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c0262408.png" /> is called strictly convex. Examples of convex functions include <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c0262409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624014.png" />. If the sign of inequality (1) is reversed, the function is called concave. All measurable convex functions on open intervals are continuous. There exist convex functions which are not continuous, but they are very irregular: If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624015.png" /> is convex on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624016.png" /> and is bounded from above on some interval lying inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624017.png" />, it is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624018.png" />. Thus, a discontinuous convex function is unbounded on any interior interval and is not measurable.
+
$$
 +
f \left( \frac{x_1 + x_2}{2} \right) \le \frac{f(x_1) + f(x_2)}{2} \tag{1}
 +
$$
  
If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624019.png" /> is continuous on an interval, and if each chord of its graph contains at least one point other than the end points of the chord and lying above the graph or on it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624020.png" /> is convex. It follows from condition (1) that for a continuous function the centre of gravity of any finite number of material points lying on the graph of the function lies either above the graph or on it: For any numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624022.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624023.png" /> is arbitrary), the [[Jensen inequality|Jensen inequality]]
+
for every two points $x_1$ and $x_2$ from this interval. The geometrical meaning of this condition is that the midpoint of any chord of the graph of the function $f$ is located either above the graph or on it. If the inequality (1) is strict for all $x_1$ and $x_2$, then $f$ is called strictly convex. Examples of convex functions include $x^p, p \ge 1$, $x \ln x$ for $x > 0$, and $\left| x \right|$ for all $x$. If the sign of inequality (1) is reversed, the function is called concave. All measurable convex functions on open intervals are continuous. There exist convex functions which are not continuous, but they are very irregular: If a function $f$ is convex on the interval $(a, b)$ and is bounded from above on some interval lying inside $(a, b)$, it is continuous on $(a, b)$. Thus, a discontinuous convex function is unbounded on any interior interval and is not measurable.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
If a function $f$ is continuous on an interval, and if each chord of its graph contains at least one point other than the end points of the chord and lying above the graph or on it, $f$ is convex. It follows from condition (1) that for a continuous function the centre of gravity of any finite number of material points lying on the graph of the function lies either above the graph or on it: For any numbers $p_k > 0, k = 1, \ldots, n$ (where $n$ is arbitrary), the [[Jensen inequality|Jensen inequality]]
 +
 
 +
$$
 +
f \left( \frac{\sum_{k = 1}^{n} p_k x_k}{\sum_{k = 1}^{n} p_k} \right) \le \frac{\sum_{k = 1}^{n} p_k f(x_k)}{\sum_{k = 1}^{n} p_k} \tag{2}
 +
$$
  
 
is valid.
 
is valid.
  
If, for some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624025.png" />, inequality (2) is true for any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624027.png" /> in some interval and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624029.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624030.png" /> is continuous and, of course, convex on this interval. Any chord of the graph of a continuous convex function coincides with the corresponding part of the graph or lies entirely above the graph except for its end points. This means that if a continuous convex function is not linear on any interval, strict inequality is realized in (1) and (2) for any pairwise different values of the argument, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624031.png" /> is a strictly convex function.
+
If, for some function $f$, inequality (2) is true for any two points $x_1$ and $x_2$ in some interval and any $p_1 > 0$ and $p_2 > 0$, the function $f$ is continuous and, of course, convex on this interval. Any chord of the graph of a continuous convex function coincides with the corresponding part of the graph or lies entirely above the graph except for its end points. This means that if a continuous convex function is not linear on any interval, strict inequality is realized in (1) and (2) for any pairwise different values of the argument, i.e. $f$ is a strictly convex function.
  
A continuous function is convex if and only if the set of points of the plane located above its graph, i.e. its [[Supergraph|supergraph]], is a [[Convex set|convex set]]. For a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624032.png" />, defined on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624033.png" />, to be convex, it is necessary and sufficient that for each point on the graph there be at least one straight line (known as a supporting line) situated under the graph (on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624034.png" />) or partly on the graph that passes through the point, i.e. for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624035.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624036.png" /> such that
+
A continuous function is convex if and only if the set of points of the plane located above its graph, i.e. its [[Supergraph|supergraph]], is a [[Convex set|convex set]]. For a continuous function $f$, defined on an interval $(a, b)$, to be convex, it is necessary and sufficient that for each point on the graph there be at least one straight line (known as a supporting line) situated under the graph (on the interval $(a, b)$) or partly on the graph that passes through the point, i.e. for any point $x_0 \in (a, b)$ there exists a $k = k(x_0)$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$
 +
f(x_0) + k(x - x_0) \le f(x) \tag{3}
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624038.png" />.
+
for all $x \in (a, b)$.
  
A continuous function which is convex on an open interval has no strict local maximum. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624039.png" /> is continuous and convex on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624040.png" />, it has, at each one of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624041.png" />, a finite left, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624042.png" />, and right, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624043.png" />, derivative; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624044.png" /> and, in addition, if the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624045.png" /> satisfies condition (3), the inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624046.png" /> hold. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624048.png" /> do not decrease, and at all points except, possibly, a countable number of them, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624049.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624050.png" /> is differentiable at these points. On each closed interval located inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624051.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624052.png" /> satisfies a Lipschitz condition and is thus absolutely continuous. This makes it possible to establish the following convexity criterion: A continuous function is convex if and only if it is the indefinite integral of a non-decreasing function.
+
A continuous function which is convex on an open interval has no strict local maximum. If a function $f$ is continuous and convex on an interval $(a, b)$, it has, at each one of its points $x_0$, a finite left, $D_{-} f(x_0)$, and right, $D_{+} f(x_0)$, derivative; moreover, $D_{-} f(x_0) \le D_{+} f(x_0)$ and, in addition, if the number $k = k(x_0)$ satisfies condition (3), the inequalities $D_{-} f(x_0) \le k(x_0) \le D_{+} f(x_0)$< hold. The functions $D_{-} f(x)$ and $D_{+} f(x)$ do not decrease, and at all points except, possibly, a countable number of them, $D_{-} f(x) = D_{+} f(x) = f'(x)$, so that $f$ is differentiable at these points. On each closed interval located inside $(a, b)$ the function $f$ satisfies a Lipschitz condition and is thus absolutely continuous. This makes it possible to establish the following convexity criterion: A continuous function is convex if and only if it is the indefinite integral of a non-decreasing function.
  
If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624053.png" /> is differentiable on an interval, it is (strictly) convex on this interval if and only if its derivative does not decrease (is increasing). At a point of the graph of a continuous convex function at which the function is differentiable there exists a unique supporting line — the tangent at this point. On the other hand, if, at any point of the graph of a function which is differentiable on an interval, the tangent to the graph at that point lies under the graph in some neighbourhood of that point (except at the tangency point itself), the function is strictly convex; if it lies under the graph or partly on it, it is just a convex function.
+
If a function $f$ is differentiable on an interval, it is (strictly) convex on this interval if and only if its derivative does not decrease (is increasing). At a point of the graph of a continuous convex function at which the function is differentiable there exists a unique supporting line — the tangent at this point. On the other hand, if, at any point of the graph of a function which is differentiable on an interval, the tangent to the graph at that point lies under the graph in some neighbourhood of that point (except at the tangency point itself), the function is strictly convex; if it lies under the graph or partly on it, it is just a convex function.
  
If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624054.png" /> is twice-differentiable on the interval, it is convex on this interval if and only if its second derivative is non-negative on this interval (this theorem is valid for the second symmetric derivative, as well as for the ordinary second derivative). If the function has a positive second derivative at each point of some interval, it is strictly convex on that interval.
+
If the function $f$ is twice-differentiable on the interval, it is convex on this interval if and only if its second derivative is non-negative on this interval (this theorem is valid for the second symmetric derivative, as well as for the ordinary second derivative). If the function has a positive second derivative at each point of some interval, it is strictly convex on that interval.
  
If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624055.png" /> are convex on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624058.png" />, then the function
+
If the functions $f_i$ are convex on an interval $(a, b)$ and $p_i > 0, i = 1, \ldots, n$, then the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624059.png" /></td> </tr></table>
+
$$
 +
f = \sum\limits_{i = 1}^{n} p_i f_i
 +
$$
  
is also convex on this interval; also, if even one of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624060.png" /> is strictly convex, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624061.png" /> is strictly convex as well.
+
is also convex on this interval; also, if even one of the functions $f_i$ is strictly convex, $f$ is strictly convex as well.
  
There exist various generalizations of the concept of convexity to functions of several variables. For instance, let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624062.png" /> be defined on a convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624063.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624064.png" />-dimensional affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624065.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624066.png" /> is called convex if inequality (1) is valid for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624068.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624069.png" /> denotes the sum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624070.png" />-dimensional vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624072.png" />. The properties of a convex function of one variable are correspondingly generalized to functions of several variables; for example, inequality (2) is satisfied only for continuous convex functions. A continuous function is convex if and only if the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624073.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624074.png" /> lying above its graph is convex.
+
There exist various generalizations of the concept of convexity to functions of several variables. For instance, let a function $y = f(x^1, \ldots, x^n)$ be defined on a convex set $M$ of the $n$-dimensional affine space $E^n$. The function $f$ is called convex if inequality (1) is valid for all points $x_1 \in M$ and $x_2 \in M$, where $x_1 + x_2$ denotes the sum of the $n$-dimensional vectors $x_1$ and $x_2$. The properties of a convex function of one variable are correspondingly generalized to functions of several variables; for example, inequality (2) is satisfied only for continuous convex functions. A continuous function is convex if and only if the set of points $(x^1, \ldots, x^n, y)$ of the space $E^{n + 1}$ lying above its graph is convex.
  
A continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624075.png" /> defined on a convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624076.png" /> is convex if and only if for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624077.png" /> there exists a linear function
+
A continuous function $f$ defined on a convex domain $G$ is convex if and only if for each point $x \in G$ there exists a linear function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624078.png" /></td> </tr></table>
+
$$
 +
l(y) = a_1 y^1 + \ldots + a_n y^n + b,
 +
$$
  
 
such that
 
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624079.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$
 +
f(x) = l(x),\qquad f(y) \ge l(y),\qquad y \in G. \tag{4}
 +
$$
  
The hyperplane defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624080.png" /> is called a supporting hyperplane.
+
The hyperplane defined by the equation $l(y) = 0$ is called a supporting hyperplane.
  
If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624081.png" /> is continuously differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624082.png" />, condition (4) is equivalent to the condition
+
If a function $f$ is continuously differentiable in $G$, condition (4) is equivalent to the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624083.png" /></td> </tr></table>
+
$$
 +
f(y) - f(x) - \sum\limits_{i = 1}^{n} \frac{\partial f(x)}{\partial x^i} (y^i - x^i) \ge 0,\qquad x, y \in G.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624084.png" /> is twice-differentiable, condition (4) is equivalent to the condition that the second differential of the function, i.e. the quadratic form
+
If $f$ is twice-differentiable, condition (4) is equivalent to the condition that the second differential of the function, i.e. the quadratic form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624085.png" /></td> </tr></table>
+
$$
 +
\sum\limits_{i = 1}^{n} \sum\limits_{j = 1}^{n} \frac{\partial f(x)}{\partial x^i \partial x^j} \xi^i \xi^j
 +
$$
  
is non-negative for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624086.png" />.
+
is non-negative for all $x \in G$.
  
 
Another important generalization of the concept of a convex function for functions of several variables is the concept of a [[Subharmonic function|subharmonic function]]. The concept of a convex function can be extended in a natural manner to include functions defined on corresponding subsets of infinite-dimensional linear spaces; cf. [[Convex functional|Convex functional]].
 
Another important generalization of the concept of a convex function for functions of several variables is the concept of a [[Subharmonic function|subharmonic function]]. The concept of a convex function can be extended in a natural manner to include functions defined on corresponding subsets of infinite-dimensional linear spaces; cf. [[Convex functional|Convex functional]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Functions of a real variable" , Addison-Wesley  (1976)  pp. Chapt. 1 Sect. 4  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)  pp. Chapt. 1</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Kudryavtsev,  "Mathematical analysis" , '''1''' , Moscow  (1973)  pp. Chapt. 1  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , Deutsch. Verlag Wissenschaft. , Frankfurt a.M.  (1961)  pp. Chapt. 10  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  pp. Chapt. 5  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1934)  pp. Chapt. 3</TD></TR></table>
 
  
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) pp. Chapt. 1 Sect. 4 (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988) pp. Chapt. 1</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.D. Kudryavtsev, "Mathematical analysis" , '''1''' , Moscow (1973) pp. Chapt. 1 (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , Deutsch. Verlag Wissenschaft. , Frankfurt a.M. (1961) pp. Chapt. 10 (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S.M. Nikol'skii, "A course of mathematical analysis" , '''1–2''' , MIR (1977) pp. Chapt. 5 (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934) pp. Chapt. 3</TD></TR></table>
  
 +
====Comments====
  
====Comments====
+
Convexity of a real-valued function $f$ on an interval $I$ is often defined by the condition that
Convexity of a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624087.png" /> on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624088.png" /> is often defined by the condition that
 
  
 +
$$
 +
f((1 - \alpha) x + \alpha y) \le (1 - \alpha) f(x) + \alpha f(y) \tag{\text{*}}
 +
$$
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624089.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624089.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
  
whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624091.png" />. This implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624092.png" /> is continuous on the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624093.png" />. For measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624094.png" />, (1) and (*) are equivalent. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026240/c02624095.png" /> satisfying (1) is also called midpoint convex.
+
whenever $x, y \in I$ and $0 \le \alpha \le 1$. This implies that $f$ is continuous on the interior of $I$. For measurable $f$, (1) and (*) are equivalent. A function $f$ satisfying (1) is also called midpoint convex.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.R. Stromberg,   "Introduction to classical real analysis" , Wadsworth (1981) pp. 199–206; 334</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V. Barbu,   Th. Precupanu,   "Convexity and optimization in Banach spaces" , Reidel (1986) pp. Chapt. 2</TD></TR></table>
+
 
 +
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 199–206; 334</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V. Barbu, Th. Precupanu, "Convexity and optimization in Banach spaces" , Reidel (1986) pp. Chapt. 2</TD></TR></table>

Revision as of 19:53, 8 July 2013

A function $f$, defined on some interval, satisfying the condition

$$ f \left( \frac{x_1 + x_2}{2} \right) \le \frac{f(x_1) + f(x_2)}{2} \tag{1} $$

for every two points $x_1$ and $x_2$ from this interval. The geometrical meaning of this condition is that the midpoint of any chord of the graph of the function $f$ is located either above the graph or on it. If the inequality (1) is strict for all $x_1$ and $x_2$, then $f$ is called strictly convex. Examples of convex functions include $x^p, p \ge 1$, $x \ln x$ for $x > 0$, and $\left| x \right|$ for all $x$. If the sign of inequality (1) is reversed, the function is called concave. All measurable convex functions on open intervals are continuous. There exist convex functions which are not continuous, but they are very irregular: If a function $f$ is convex on the interval $(a, b)$ and is bounded from above on some interval lying inside $(a, b)$, it is continuous on $(a, b)$. Thus, a discontinuous convex function is unbounded on any interior interval and is not measurable.

If a function $f$ is continuous on an interval, and if each chord of its graph contains at least one point other than the end points of the chord and lying above the graph or on it, $f$ is convex. It follows from condition (1) that for a continuous function the centre of gravity of any finite number of material points lying on the graph of the function lies either above the graph or on it: For any numbers $p_k > 0, k = 1, \ldots, n$ (where $n$ is arbitrary), the Jensen inequality

$$ f \left( \frac{\sum_{k = 1}^{n} p_k x_k}{\sum_{k = 1}^{n} p_k} \right) \le \frac{\sum_{k = 1}^{n} p_k f(x_k)}{\sum_{k = 1}^{n} p_k} \tag{2} $$

is valid.

If, for some function $f$, inequality (2) is true for any two points $x_1$ and $x_2$ in some interval and any $p_1 > 0$ and $p_2 > 0$, the function $f$ is continuous and, of course, convex on this interval. Any chord of the graph of a continuous convex function coincides with the corresponding part of the graph or lies entirely above the graph except for its end points. This means that if a continuous convex function is not linear on any interval, strict inequality is realized in (1) and (2) for any pairwise different values of the argument, i.e. $f$ is a strictly convex function.

A continuous function is convex if and only if the set of points of the plane located above its graph, i.e. its supergraph, is a convex set. For a continuous function $f$, defined on an interval $(a, b)$, to be convex, it is necessary and sufficient that for each point on the graph there be at least one straight line (known as a supporting line) situated under the graph (on the interval $(a, b)$) or partly on the graph that passes through the point, i.e. for any point $x_0 \in (a, b)$ there exists a $k = k(x_0)$ such that

$$ f(x_0) + k(x - x_0) \le f(x) \tag{3} $$

for all $x \in (a, b)$.

A continuous function which is convex on an open interval has no strict local maximum. If a function $f$ is continuous and convex on an interval $(a, b)$, it has, at each one of its points $x_0$, a finite left, $D_{-} f(x_0)$, and right, $D_{+} f(x_0)$, derivative; moreover, $D_{-} f(x_0) \le D_{+} f(x_0)$ and, in addition, if the number $k = k(x_0)$ satisfies condition (3), the inequalities $D_{-} f(x_0) \le k(x_0) \le D_{+} f(x_0)$< hold. The functions $D_{-} f(x)$ and $D_{+} f(x)$ do not decrease, and at all points except, possibly, a countable number of them, $D_{-} f(x) = D_{+} f(x) = f'(x)$, so that $f$ is differentiable at these points. On each closed interval located inside $(a, b)$ the function $f$ satisfies a Lipschitz condition and is thus absolutely continuous. This makes it possible to establish the following convexity criterion: A continuous function is convex if and only if it is the indefinite integral of a non-decreasing function.

If a function $f$ is differentiable on an interval, it is (strictly) convex on this interval if and only if its derivative does not decrease (is increasing). At a point of the graph of a continuous convex function at which the function is differentiable there exists a unique supporting line — the tangent at this point. On the other hand, if, at any point of the graph of a function which is differentiable on an interval, the tangent to the graph at that point lies under the graph in some neighbourhood of that point (except at the tangency point itself), the function is strictly convex; if it lies under the graph or partly on it, it is just a convex function.

If the function $f$ is twice-differentiable on the interval, it is convex on this interval if and only if its second derivative is non-negative on this interval (this theorem is valid for the second symmetric derivative, as well as for the ordinary second derivative). If the function has a positive second derivative at each point of some interval, it is strictly convex on that interval.

If the functions $f_i$ are convex on an interval $(a, b)$ and $p_i > 0, i = 1, \ldots, n$, then the function

$$ f = \sum\limits_{i = 1}^{n} p_i f_i $$

is also convex on this interval; also, if even one of the functions $f_i$ is strictly convex, $f$ is strictly convex as well.

There exist various generalizations of the concept of convexity to functions of several variables. For instance, let a function $y = f(x^1, \ldots, x^n)$ be defined on a convex set $M$ of the $n$-dimensional affine space $E^n$. The function $f$ is called convex if inequality (1) is valid for all points $x_1 \in M$ and $x_2 \in M$, where $x_1 + x_2$ denotes the sum of the $n$-dimensional vectors $x_1$ and $x_2$. The properties of a convex function of one variable are correspondingly generalized to functions of several variables; for example, inequality (2) is satisfied only for continuous convex functions. A continuous function is convex if and only if the set of points $(x^1, \ldots, x^n, y)$ of the space $E^{n + 1}$ lying above its graph is convex.

A continuous function $f$ defined on a convex domain $G$ is convex if and only if for each point $x \in G$ there exists a linear function

$$ l(y) = a_1 y^1 + \ldots + a_n y^n + b, $$

such that

$$ f(x) = l(x),\qquad f(y) \ge l(y),\qquad y \in G. \tag{4} $$

The hyperplane defined by the equation $l(y) = 0$ is called a supporting hyperplane.

If a function $f$ is continuously differentiable in $G$, condition (4) is equivalent to the condition

$$ f(y) - f(x) - \sum\limits_{i = 1}^{n} \frac{\partial f(x)}{\partial x^i} (y^i - x^i) \ge 0,\qquad x, y \in G. $$

If $f$ is twice-differentiable, condition (4) is equivalent to the condition that the second differential of the function, i.e. the quadratic form

$$ \sum\limits_{i = 1}^{n} \sum\limits_{j = 1}^{n} \frac{\partial f(x)}{\partial x^i \partial x^j} \xi^i \xi^j $$

is non-negative for all $x \in G$.

Another important generalization of the concept of a convex function for functions of several variables is the concept of a subharmonic function. The concept of a convex function can be extended in a natural manner to include functions defined on corresponding subsets of infinite-dimensional linear spaces; cf. Convex functional.

References

[1] N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) pp. Chapt. 1 Sect. 4 (Translated from French)
[2] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) pp. Chapt. 1
[3] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) pp. Chapt. 1 (In Russian)
[4] I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , Deutsch. Verlag Wissenschaft. , Frankfurt a.M. (1961) pp. Chapt. 10 (Translated from Russian)
[5] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) pp. Chapt. 5 (Translated from Russian)
[6] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934) pp. Chapt. 3

Comments

Convexity of a real-valued function $f$ on an interval $I$ is often defined by the condition that

$$ f((1 - \alpha) x + \alpha y) \le (1 - \alpha) f(x) + \alpha f(y) \tag{\text{*}} $$

(*)

whenever $x, y \in I$ and $0 \le \alpha \le 1$. This implies that $f$ is continuous on the interior of $I$. For measurable $f$, (1) and (*) are equivalent. A function $f$ satisfying (1) is also called midpoint convex.

References

[a1] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 199–206; 334
[a2] V. Barbu, Th. Precupanu, "Convexity and optimization in Banach spaces" , Reidel (1986) pp. Chapt. 2
How to Cite This Entry:
Convex function (of a real variable). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convex_function_(of_a_real_variable)&oldid=29900
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article