In order for convenience in checking definition along Learning in Math, I found it necessary to gather definitions of the same kind to distinguish the differences among them. This work is very hard to do even many materials are available; so if you want to use for commercial use, please contact me for permission, otherwise. Furthermore, the work is still long from finished, if you want to contribute for more or better definitions, please contact me.

Tianyu Zhang

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### 1. Separate

#### Definition

#### Competitive Definition

#### Reference

Convex Analysis, Rockafellar RT, Chapter 3 Section 11(Definition)

Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis Chapter I(Competitive Definition)

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

## 2. Properly Separate

#### Definition

#### Reference

Convex Analysis, Rockafellar RT, Chapter 3 Section 11

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

## 3. Strongly Separate

#### Definition

#### Remark

Not every pair of disjoint closed convex sets can be separated strongly.

#### Reference

Convex Analysis, Rockafellar RT, Chapter 3 Section 11

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

## 4. Strictly Separate

#### Definition

#### Competitive Definition

#### Reference

Convex Analysis, Rockafellar RT, Chapter 3 Section 11(Definition)

Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis Chapter I(Competitive Definition)

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas