Math 2.0

Real functions Definition

Strictly convex 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 strictly convex if f(tx+(1−t)y)<tf(x)+(1−t)f(y)f(t x + (1 - t) y) < t f(x) + (1 - t) f(y) for all x,y∈Ix, y \in I with x≠yx \ne y and all t∈(0,1)t \in (0, 1).

Remarks

Each strictly convex function is convex. A linear function is convex but not strictly convex. A strictly convex function has at most one minimum point. For example, x2x^2 and exe^x are strictly convex. Mathlib defines strict convexity as StrictConvexOn.

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Further reading

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