Definition of concavity

Question

The mathematical definition of a function being concave between points $x_1$ and $x_2$ is the following:

$\lambda f(x_1)+(1-\lambda)f(x_2) \leq f(\lambda x_1+(1-\lambda)x_2)$, for any $0 \leq \lambda \leq 1$.

Can someone give a detailed, intuitive explanation of this theorem?

Another definition, that I think is more intuitive or fundamental, is that a function is convex iff its epigraph is convex. This explains why the word "convex", already used to describe sets, is also used to describe functions. — littleO, Jan 10 '14 at 01:00

score 1 · Accepted Answer · answered Jan 10 '14 at 01:03

1

Consider the two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$. The function $f$ between $x_1$ and $x_2$ is no lower than the line segment connecting these two points. For example, $\lambda=0.5$ corresponds to the halfway point. The LHS corresponds to the midpoint of the line segment, the RHS corresponds to the function evaluated at the average of $x_1$ and $x_2$.

Note: you have a typo in your statement, it should begin $\lambda f(x_1)$ not $\lambda(x_1)$.

answered Jan 10 '14 at 01:03

vadim123

82,796

I misspoke -- I meant concave, so I will change the question to that. – fool Jan 10 '14 at 01:10
But that doesn't change the answer, except to put the function below the line instead of above it. Do you need further explanation? – Betty Mock Jan 10 '14 at 05:14
@BettyMock No further explanation needed; thanks. – fool Jan 12 '14 at 04:59

Definition of concavity

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