# Category:Convex Sets (Vector Spaces)

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This category contains results about Convex Sets (Vector Spaces) in the context of Vector Spaces.

Definitions specific to this category can be found in Definitions/Convex Sets (Vector Spaces).

### Definition 1

We say that $C$ is **convex** if and only if:

- $t x + \paren {1 - t} y \in C$

for each $x, y \in C$ and $t \in \closedint 0 1$.

### Definition 2

We say that $C$ is **convex** if and only if:

- $t C + \paren {1 - t} C \subseteq C$

for each $t \in \closedint 0 1$, where $t C + \paren {1 - t} C$ denotes a linear combination of subsets.

## Subcategories

This category has the following 9 subcategories, out of 9 total.

## Pages in category "Convex Sets (Vector Spaces)"

The following 29 pages are in this category, out of 29 total.

### C

- Closed Convex Set in terms of Bounded Linear Functionals
- Closed Unit Ball is Convex Set
- Closure of Convex Set in Topological Vector Space is Convex
- Closure of Convex Subset in Normed Vector Space is Convex
- Convex Hull is Smallest Convex Set containing Set
- Convex Open Neighborhood of Origin in Topological Vector Space contains Balanced Convex Open Neighborhood
- Convex Set is Contractible
- Convex Set is Simply Connected
- Convex Set is Star Convex Set