Convex function: Difference between revisions

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The second statement characterizing convex functions that are valued in the real line <math>\R</math> is also the statement used to define '''{{em|convex functions}}''' that are valued in the [[extended real number line]] <math>[-\infty, \infty] = \R \cup \{\pm\infty\},</math> where such a function <math>f</math> is allowed to (but is not required to) take <math>\pm\infty</math> as a value. The first statement is not used because it permits <math>t</math> to take <math>0</math> or <math>1</math> as a value, in which case, if <math>f\left(x_1\right) = \pm\infty</math> or <math>f\left(x_2\right) = \pm\infty,</math> respectively, then <math>t f\left(x_1\right) + (1 - t) f\left(x_2\right)</math> would be undefined (because the multiplications <math>0 \cdot \infty</math> and <math>0 \cdot (-\infty)</math> are undefined). The sum <math>-\infty + \infty</math> is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of <math>-\infty</math> and <math>+\infty</math> as a value.

The second statement can also be modified to get the definition of {{em|strict convexity}}, where the latter is obtained by replacing <math>\,\leq\,</math> with the strict inequality <math>\,<.</math>

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A strictly convex function <math>f</math> is a function that the straight line between any pair of points on the curve <math>f</math> is above the curve <math>f</math> except for the intersection points between the straight line and the curve.

The function <math>f</math> is said to be '''{{em|[[Concave function|concave]]}}''' (resp. '''{{em|strictly concave}}''') if <math>-f</math> (<math>f</math> multiplied by -1−1) is convex (resp. strictly convex ).

==Alternative naming==