# Equivalence of Definitions of Convex Set in Vector Space

## Theorem

Let $\Bbb F \in \set {\R, \C}$.

Let $V$ be a vector space over $\Bbb F$.

Let $C \subseteq V$.

The following definitions of the concept of convexity are equivalent:

### 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.

## Proof

From the definition of the linear combination $t C + \paren {1 - t} C$, we have:

$t C + \paren {1 - t} C = \set {t x + \paren {1 - t} y : x, y \in C}$

for $t \in \R$.

So we have:

$t C + \paren {1 - t} C \subseteq C$
$t x + \paren {1 - t} y \in C$ for all $x, y \in C$ and $t \in \closedint 0 1$.

Hence the result.

$\blacksquare$