We consider a **normed vector space** \(E\) over the field of the reals \(\mathbb{R}\) and a **convex** subset \(C \subset E\).

We suppose that \(0 \in C\) and that \(C\) is **unbounded**, i.e. there exists points in \(C\) at distance as big as we wish from \(0\).

The following question arises: “*does \(C\) contains a ray*?”. It turns out that the answer depends on the

**dimension**of the space \(E\). If \(E\) is of

**finite dimension**, then \(C\) always contains a ray, while if \(E\) is of

**infinite dimension**\(C\) may not contain a ray.

## Finite-dimensional case

Even if it means reducing the dimension of the space, we can suppose that \(0\) is an interior point of \(C\). By hypothesis, there exists a sequence \((y_n)_{n \in \mathbb{N}}\) with \(y_n \in C\) and \(\Vert y_n \Vert > n \) for all \(n \in \mathbb{N}\). The sequence \((\frac{y_n}{\Vert y_n \Vert})_{n \in \mathbb{N}}\) belongs to the unit sphere which is **compact** as \(E\) is supposed finite-dimensional. Therefore it has a converging subsequence (this is the Bolzano-Weierstrass theorem). By renumbering the sequence \((y_n)_{n \in \mathbb{N}}\), we can even suppose that \((\frac{y_n}{\Vert y_n \Vert})_{n \in \mathbb{N}}\) itself converge to a vector \(v\).

Let’s prove that \(C\) contains points as far as we want from \(0\) on the line \(\mathbb{R} v\). We pick-up a real \(A > 0\). There exists \(N \in \mathbb{N}\) such that \(\Vert y_n \Vert \ge A\) for all \(n \ge N\). As \(0 \in C\) and \(C\) is supposed convex, the points \(w_n = A \frac{y_n}{\Vert y_n \Vert}\) belongs to \(C\) for \(n \ge N\). Moreover the sequence \((w_n)_{n \in \mathbb{N}}\) converges to \(A v\). As \(0\) is supposed to be an interior point of \(C\), \(w_n – Av \in C\) for \(n\) big enough. Then also:

\[\frac{1}{2}A v= \frac{1}{2}(w_n+(A v -w_n)) \in C\] Finally, the ray \([0,+\infty v)\) belongs to \(C\).

## A counter-example in the infinite-dimensional case

Let’s take a sequence \(\textbf{e}=(e_n)_{n \in \mathbb{N}}\) of linearly independent vectors whose sequence of norms converges to \(+ \infty\) and for \(C\) the **convex hull** of that sequence. If \(C\) contains a ray, it also contains two points \(a\) and \(b\) that are **convex combinations** of vectors of the sequence \(\textbf{e}\). Therefore, \(a\) and \(b\) belongs to a finite-dimensional space \(F=\mathbb{R}e_1 + \cdots + \mathbb{R}e_N\) as well as the ray that is supposed to be contained in \(C\). But \(C \cap F\), which is the convex hull of \(\{e_1, \dots, e_n\}\) is compact (see here), a contradiction.