Equivalence of Definitions of Convex Set in Vector Space
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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.
$\blacksquare$