Convex Cone is Convex Set
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Theorem
Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $P \subseteq X$ be a convex cone in $X$.
Then $P$ is convex.
Proof
Let $x, y \in P$.
Let $t \in \closedint 0 1$ so that:
- $t \ge 0$ and $1 - t \ge 0$.
Since $P$ is a cone, we have:
- $t x \in P$ and $\paren {1 - t} y \in P$.
Since $P$ is a convex cone, we have:
- $t x + \paren {1 - t} y \in P$
So $P$ is convex.
$\blacksquare$
Sources
- 2023: Jean-Bernard Bru and Walter Alberto de Siqueira Pedra: C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics ... (previous) ... (next): $1.1$: Basic notions