Convex set with empty interior is nowhere dense?

Question

Suppose $C\subseteq\mathbb R^n$ is a convex set and $C^o=\varnothing$. Is it necessarily true that $(\overline C)^o=\varnothing$? In general, is this true if $\mathbb R^n$ is replaced by a topological vector space $X$? Or can a counterexample be found?

I know that if $C^o\neq\varnothing$, then $C^o=(\overline C)^o$, so my question is whether this result can be generalized to the case when $C^o=\varnothing$.

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Update: It's not true in general topological vector spaces. Indeed, if $X$ is a topological vector space and $Y$ is a proper dense subspace (such examples can be constructed), then $Y$ is convex and has empty interior, but $(\overline Y)^o=X^o=X$ is clearly not empty. Yet, I'm still not sure if the result holds for finite-dimensional vector spaces (in which every proper subspace is closed and thus no proper subspace is dense, so the preceding counterexample doesn't work).

Answer 1

This is true in $\mathbb{R}^n$. Say $n = 3$. If $\overline{C}$ has non empty interior that $C$ is dense in some ball hence it contains the vertices of a tetrahedron (just take a tetrahedron inside the ball and move its vertices slightly to land on a point in $C$). But now by convexity, $C$ must contain the whole solid tetrahedron which contradicts the fact that $C$ has empty interior. In higher dimensions, you can start with a generalized cube and move its vertices slightly to enter $C$.

Answer 2

Here is a rigorous—and, accordingly, a bit lengthy—operationalization of the intuition offered by the answer by @OohAah.

$\textbf{Proposition}\phantom{---}$ Let $C\subseteq \mathbb R^n$ ($n\in\mathbb N$) be a convex set. If $C$ has no interior, then nor does $\overline C$.

$\textit{Proof}\phantom{---}$ Suppose that $(\overline C)^o\neq\varnothing$. I will show that $C^o$ is not empty, either. Let $\mathbf x_0\in (\overline C)^o$. Then, there exists some $\varepsilon>0$ such that $\mathbf x_0\in B(2\varepsilon,\mathbf x_0)\subseteq\overline C$, where $B(2\varepsilon,\mathbf x_0)$ denotes the open ball of radius $2\varepsilon$ around $\mathbf x_0$ with respect to the Euclidean norm. Let $\{\mathbf e_i\}_{i=1}^n$ denote the standard basis of $\mathbb R^n$ and define $\mathbf x_i\equiv \mathbf x_0+\varepsilon \mathbf e_i$ for each $i\in\{1,\ldots,n\}$. Clearly, $$\{\mathbf x_i\}_{i=0}^n\subseteq B(2\varepsilon,\mathbf x_0)\subseteq \overline C.$$ That is, $\{\mathbf x_i\}_{i=0}^n$ is included in the closure of $C$, so that, for each $i\in\{0,1,\ldots,n\}$, there exists some $\mathbf y_i$ such that

• $\mathbf y_i\in C$; and
• $\mathbf y_i\in B((2\sqrt n)^{-1}\varepsilon,\mathbf x_i)$.

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For each $i\in\{1,\ldots,n\}$ define $\mathbf b_i\equiv \mathbf y_i-\mathbf y_0$. I claim that the vectors $\{\mathbf b_i\}_{i=1}^n$ are linearly independent. Indeed, suppose that $\lambda_1,\ldots,\lambda_n\in\mathbb R$ satisfy $$\sum_{i=1}^n\lambda_i\mathbf b_i=0.$$ Suppose, for the sake of contradiction, that not all of $\{\lambda_i\}_{i=1}^n$ are zero. Then, $$\sum_{i=1}^n|\lambda_i|>0\tag{1}.$$ In addition, \begin{align*} 0=&\,\sum_{i=1}^n\lambda_i\mathbf b_i=\sum_{i=1}^n\lambda_i(\mathbf y_i-\mathbf y_0)=\sum_{i=1}^n\lambda_i(\mathbf y_i-\mathbf x_i+\mathbf x_i-\mathbf x_0+\mathbf x_0-\mathbf y_0)\\=&\,\sum_{i=1}^n\lambda_i(\mathbf y_i-\mathbf x_i+\varepsilon\mathbf e_i+\mathbf x_0-\mathbf y_0), \end{align*} or \begin{align*} -\varepsilon\sum_{i=1}^n\lambda_i\mathbf e_i=\sum_{i=1}^n\lambda_i(\mathbf y_i-\mathbf x_i+\mathbf x_0-\mathbf y_0). \end{align*} Clearly, the Euclidean norm of the left-hand side is $\varepsilon\sqrt{\sum_{i=1}^n\lambda_i^2}$, so the following chain of inequalities holds true: \begin{align*} &\varepsilon\sqrt{\sum_{i=1}^n\lambda_i^2}=\left\|\sum_{i=1}^n\lambda_i(\mathbf y_i-\mathbf x_i+\mathbf x_0-\mathbf y_0)\right\|\leq\sum_{i=1}^n|\lambda_i|\left(\|\mathbf y_i-\mathbf x_i\|+\|\mathbf x_0-\mathbf y_0\|\right)\\ \underset{\text{see (1)}}{<}&\,\sum_{i=1}^n|\lambda_i|\left(\frac{\varepsilon}{2\sqrt{n}}+\frac{\varepsilon}{2\sqrt{n}}\right)=\sum_{i=1}^n|\lambda_i|\frac{\varepsilon}{\sqrt{n}}\leq\sqrt{\sum_{i=1}^n|\lambda_i|^2}\sqrt{\sum_{i=1}^n\frac{\varepsilon^2}{n}}=\varepsilon\sqrt{\sum_{i=1}^n\lambda_i^2}, \end{align*} where I used the Cauchy–Schwarz inequality. This is a contradiction, which implies that $\lambda_1=\ldots=\lambda_n=0$. Hence, the vectors $\{\mathbf b_i\}_{i=1}^n$ are linearly independent, and since $\dim\mathbb R^n=n$, it follows also that they constitute a basis.

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If $\mathbf z\in\mathbb R^n$, there exists a unique $(\mu_i)_{i=1}^n\in\mathbb R^n$ such that $\mathbf z=\sum_{i=1}^n\mu_i\mathbf b_i$. Define $\|\mathbf z\|_b\equiv\sum_{i=1}^n|\mu_i|$. Then, $\|\cdot\|_b$ is a norm and since $\mathbb R^n$ is finite-dimensional, it must be equivalent to the Euclidean norm. Therefore, there exists some $\xi_b>0$ such that $\|\cdot\|_b\leq \xi_b\|\cdot\|$.

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Let $$D\equiv\left\{\sum_{i=0}^n\alpha_i\mathbf y_i\,\Bigg|\,\alpha_i\geq0\text{ for all $i\in\{0,1,\ldots,n\}$ and }\sum_{i=0}^n\alpha_i=1\right\}.$$ Since $\{\mathbf y_i\}_{i=0}^n\subseteq C$ and $C$ is convex, it follows that $D\subseteq C$. Define $$\mathbf w\equiv\sum_{i=0}^n\frac{1}{n+1}\mathbf y_i.$$ Clearly, $\mathbf w\in D$ and $$\mathbf w-\mathbf y_0=\sum_{i=1}^{n}\frac{1}{n+1}(\mathbf y_i-\mathbf y_0)=\sum_{i=1}^{n}\frac{1}{n+1}\mathbf b_i.$$

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Let $$\delta\equiv\frac{1}{n(n+1)\xi_b}>0.$$ I will show that $B(\delta,\mathbf w)\subseteq D$. To this end, pick any $\mathbf z\in B(\delta,\mathbf w)$, so that $\|\mathbf z-\mathbf w\|<\delta.$ Since $\{\mathbf b_i\}_{i=1}^n$ is a basis, there exists a unique $(\mu_i)_{i=1}^n\in\mathbb R^n$ such that $\mathbf z-\mathbf y_0=\sum_{i=1}^n\mu_i\mathbf b_i$. Then, $$\mathbf z-\mathbf w=(\mathbf z-\mathbf y_0)-(\mathbf w-\mathbf y_0)=\sum_{i=1}^n\left(\mu_i-\frac{1}{n+1}\right)\mathbf b_i.$$ Consequently, for any $i\in\{1,\ldots,n\}$, it follows that \begin{align*} \left|\mu_i-\frac{1}{n+1}\right|\leq\sum_{j=1}^n\left|\mu_j-\frac{1}{n+1}\right|=\|\mathbf z-\mathbf w\|_b\leq \xi_b\|\mathbf z-\mathbf w\|<\xi_b\delta=\frac{1}{n(n+1)}. \end{align*} Hence, $$0\leq\frac{(n-1)}{n(n+1)}=\frac{1}{n+1}-\frac{1}{n(n+1)}<\mu_i<\frac{1}{n+1}+\frac{1}{n(n+1)}=\frac{1}{n},$$ for each $i\in\{1,\ldots,n\}$ so that $\sum_{i=1}^n\mu_i<1$. It follows that \begin{align*} \mathbf z=\mathbf y_0+\sum_{i=1}^n\mu_i\mathbf b_i=\mathbf y_0+\sum_{i=1}^n\mu_i(\mathbf y_i-\mathbf y_0)=\left(1-\sum_{i=1}^n\mu_i\right)\mathbf y_0+\sum_{i=1}^n\mu_i\mathbf y_i, \end{align*} so $\mathbf z\in D$.

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Conclusion:

• $\mathbf w\in B(\delta,\mathbf w)\subseteq D\subseteq C$; and
• $B(\delta,\mathbf w)$ is a non-empty open set; so that
• $\mathbf w\in C^o$.

In particular, $C^o$ is not empty. $\blacksquare$

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Intuitively, the points $\{\mathbf x_i\}_{i=0}^n$ in $\overline C$ span an $n$-dimensional simplex, whose vertices are perturbed slightly so that the new vertices do not lie in the same hyperplane and are all in $C$. The convex hull of these new vertices gives rise to a “distorted simplex” fully contained in $C$. Then, the centroid of this distorted simplex can be surrounded by a small ball still included in the distorted simplex, and hence in $C$.