Smooth Convex Bodies in $\mathbb{R}^n$ with Dense Union of Facets
Keywords:
convex body, Euclidean space, exposed point, Grassmann manifoldAbstract
Let $B$ be closed and convex in $\mathbb{R}^n$; $B$ is called a convex body if $B$ is compact and has a nonempty interior with respect to $\mathbb{R}^n$. In addition, $B$ is smooth if $B$ has a unique supporting hyperplane at every boundary point. Let $k, n \in \mathbb{N}$ with $k < n$ and let $\mathbb{L}^n_k$ denote the Grassmann manifold consisting of all $k$-dimensional linear subspaces in $\mathbb{R}^n$. An intersection $F$ of $B$ and a supporting hyperplane is called a facet if $\dim F = n - 1$. A point $x$ of $B$ is called exposed by $\mathcal{P} \subset \mathbb{L}^n_k$ if there is a $P \in \mathcal{P}$ such that $(x + P) \cap B = \{x\}$. In this paper, for every $n \geq 2$, we have constructed symmetric smooth convex bodies $B(n)$ in $\mathbb{R}^n$ whose union of all facets is dense in the boundary of $B(n)$ and so that the set of its facets defines a dense set $P$ in $\mathbb{L}^n_k$ such that the set of all points in $B(n)$ exposed by $P$ is empty.
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