Linear Closure Equals Topological Closure for Solid Convex Sets

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

Let $X$ be a normed vector space and let $\Omega\subset X$ be convex, with int(Ω) $\neq\emptyset$ .

Proposition:

\operatorname{lin}(\Omega)=\overline{\Omega},

where $\overline{\Omega}$ denotes the closure of $\Omega$ in the norm-induced topology.

Remarks

Context: This result says that, for "solid" convex sets, the algebraic construction lin(Ω) (built from segments) recovers the usual topological closure. Compare with core equals interior for the analogous result on the interior side.