Logarithmic convexity

A logarithmically convex function is a positive function for which the concatenation of the function with the logarithm is convex . Logarithmic convexity of functions is a special case of convexity of functions and plays a role in the characterization of the gamma function by means of Bohr-Mollerup's uniqueness theorem and in variants of convex optimization . ${\displaystyle f}$

definition

Let there be a function with and for everyone . Then is called ${\displaystyle f:D\mapsto \mathbb {R} }$ ${\displaystyle D\subset \mathbb {R} ^{n}}$ ${\displaystyle f(x)>0}$ ${\displaystyle x\in D}$ ${\displaystyle f}$

• logarithmic convex when is convex. ${\displaystyle \log(f(x))}$
• logarithmic concave when is concave . ${\displaystyle \log(f(x))}$

If it is a convex set , it is equivalent to ${\displaystyle D}$

• ${\displaystyle f}$is logarithmically convex if and only if for all and all that ${\displaystyle x,y\in D}$ ${\displaystyle \lambda \in [0,1]}$

${\displaystyle f(\lambda x+(1-\lambda )y)\leq f(x)^{\lambda }f(y)^{1-\lambda }}$.

• ${\displaystyle f}$is logarithmically concave if and only if for all and all that ${\displaystyle x,y\in D}$ ${\displaystyle \lambda \in [0,1]}$

${\displaystyle f(\lambda x+(1-\lambda )y)\geq f(x)^{\lambda }f(y)^{1-\lambda }}$.

Logarithmic convexity can also be defined for functions , then you have to fall back on an extended definition of convexity of functions that also covers the function values . ${\displaystyle f(x)\geq 0}$ ${\displaystyle \pm \infty }$

Examples

• Since the composition of convex functions , is convex if g monotonically increasing , and the exponential function is both convex monotonically increasing, convex functions are logarithmic also convex. The reverse is generally not true. For example, is a convex function, but is not convex. Therefore is convex but not logarithmically convex. ${\displaystyle g\circ f}$ ${\displaystyle f(x)=x^{2}}$ ${\displaystyle \log f(x)=\log x^{2}=2\log |x|}$ ${\displaystyle f(x)=x^{2}}$
• A particularly important example of a logarithmically convex function is the gamma function. According to Bohr-Mollerup's theorem , every logarithmically convex function auf that satisfies the functional equation is a multiple of the gamma function. ${\displaystyle (0,\infty )}$ ${\displaystyle f(x+1)=xf(x)}$
• Some important probability densities are logarithmically concave, for example those of the Gaussian distribution and the exponential distribution .

properties

• A function is logarithmically convex if and only if it is logarithmically concave and vice versa. ${\displaystyle f}$ ${\displaystyle {\frac {1}{f}}}$
• Products of logarithmically convex (concave) functions are again logarithmically convex (concave).
• The sum of two logarithmically convex functions is again logarithmically convex. The analogous statement for logarithmically concave functions does not generally apply.
• If one defines , then logarithmic convexity can also be defined for functions that take on the value . A function is logarithmically convex (concave) if the extended function is convex (concave) as an extended function. ${\displaystyle \log(0):=-\infty }$ ${\displaystyle 0}$ ${\displaystyle \log(f(x))}$
• Since every logarithmic convex function is convex, it is also always quasi-convex .
• Furthermore, logarithmically convex functions take over all properties of convex functions, in particular all sub-level sets and the epigraph of a logarithmically convex function are convex sets.

literature

• Stephen Boyd, Lieven Vandenberghe: Convex Optimization . Cambridge University Press, Cambridge, New York, Melbourne 2004, ISBN 978-0-521-83378-3 , pp. 104-108 ( online ).