Convex Cone is Convex Set

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