What are log-concave functions?

Functions tend to be logarithmically concave if they follow a particular set of rules and conditions. In this Answer, we'll understand what constitutes such functions and examples that can elaborate the log-concave functions in a much simpler way.

Logarithmically concave functions

A function is log-concave if $log$ f is concave:

Properties and identities

The properties of concave functions are listed as follows:

• Log-concave functions are also quasi-convex.Identity 1
• Twice differentiable functions with a convex domain should satisfy the following condition:

  For a single variable, the condition can be defined as follows:

• All concave functions with a positive domain are log-concave. However, a function that might not be concave in its derivative form may end up not being log-concave due to a violation of the conditions.
• The product of log-concave functions is always log-concave.
• The sum of log-concave functions may not be log-concave.
• If the function $f$ that casts $R^m \times R^n$ to $R$ is log-concave, then the following equation is also log-concave:

Now that we know about the properties of log-concave functions in detail, let's draw the graph of a log-concave function and check if it fulfills the conditions.

Examples

A few examples of log-concave functions are listed as follows:

• $x^b$ is log-convex for $b \le 0$, but log-concave for $b \ge 0$
• Many probability densities are log-concave such as Laplace:

• Uni variate parametric families are log-concave such as:

• The cumulative Gaussian distribution is log-concave:

• If $K \sub R^d$ is compact and convex, then $f(x) = 1_K(x)/\lambda(K)$ is a log-concave density.