Math 2.0

Real functions Definition

Concave function

Builds on Convex function

Definition

Let II be an interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be a function. The function ff is concave if f(tx+(1−t)y)≥tf(x)+(1−t)f(y)f(t x + (1 - t) y) \ge t f(x) + (1 - t) f(y) for all x,y∈Ix, y \in I and all t∈[0,1]t \in [0, 1].

Remarks

So ff is concave exactly when −f-f is convex. For example, x\sqrt{x} and ln⁡x\ln x are concave on (0,∞)(0, \infty). Mathlib defines concavity on a set as ConcaveOn.

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  1. Convex function definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Concave_function, Definition
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