Convex set

Builds on Euclidean space Affine combination

Definition

A subset $S$ of $\mathbb{R}^n$ is convex if for all $x, y \in S$ and each real number $t$ with $0 \leq t \leq 1$ , $(1 - t)\, x + t\, y \in S.$ That is, $S$ contains the segment from $x$ to $y$ .

Remarks

The same definition works in each real vector space or real affine space. Mathlib's Convex is stated for a module over an ordered semiring. A convex set contains each convex combination $\sum_i w_i x_i$ of its points, an affine combination with $w_i \geq 0$ . The empty set and the whole space are convex. Each intersection of convex sets is convex, but a union of convex sets need not be convex. Balls, half-spaces, lines and triangles with their inside are convex.

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Affine combination concept
Euclidean space object
Convex set definition

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3 pages build on this one.

Tag	`0K06X9QXDX`
URI	`m2://0K06X9QXDX`
Kind	definition
Writing	W1 stated
MSC	52A20
Defines	Convex set
Uses	Euclidean space, Affine combination

Convex set

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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