The set of boundary vectors of compact convex body has empty interior

Question

Let $K$ be a compact convex body in the Euclidean space $\mathbb R^n$ and $\partial K$ be its topological boundary in $\mathbb R^n$.

Definition. A vector $\mathbf v\in\mathbb R^n$ is called $K$-boundary if there exists a point $\mathbf a\in\partial K$ such that $[\mathbf a,\mathbf a+\varepsilon\mathbf v]\subseteq\partial K$ for some $\varepsilon>0$.

Conjecture: The set of $K$-boundary vectors has empty interior in $\mathbb R^n$.
More precisely, this set has Hausdorff dimension $\le n-1$.

Question. Is this conjecture true? And if yes, where is it proved?
It sounds like a basic fact in Convexity Theory.

Remark 2. For $n=2$, Conjecture is true: the set of $K$-boundary vectors coincides with a countable union of linear subspaces of dimension $\le 1$ and hence it has empty interior in $\mathbb R^2$. Moreover, it is meager and Lebesgue null in $\mathbb R^2$.

Remark 3. For $n=3$, Conjecture also is true: the set of $K$-boundary vectors has Hausdorff dimension 2 and hence is Lebesgue null and meager in $\mathbb R^3$, see my partial answer below.

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Remark $n$ (Added in $n$-th Edit): It seems that the 3-dimensional agrument used in my answer below easily generalizes to higher dimensions and proves

Theorem. For any compact body $K$ in a Euclidean space $\mathbb R^n$ and any $k\le n$ the subspace $$\big\{L\in G(k,\mathbb R^n):\exists x\in\partial K\;\dim(\partial K\cap(x+L))=k\big\}$$of the Grassmannian $Gr(k,\mathbb R^n)$ has Hausdorff dimension $n-k-1$.

So, now the problem is to find this theorem somewhere in literature (I tend to believe that such a result had to appear somewhere).

Answer 1

Here is a proof of Conjecture in dimension $n=3$.

Let $K$ be a compact convex body in $\mathbb R^3$. For $k\le 3$, let $Gr(k,3)$ be the space of $k$-dimensional linear subspaces of $\mathbb R^3$. So, $Gr(1,3)$ is the space of lines and $Gr(2,3)$ is the space of planes in $\mathbb R^3$.

A point $x\in\partial K$ will be called

$\bullet$ flat if there is a plane $P\in Gr(2,3)$ such that $x$ is an interior point of the intersection $\partial K\cap(x+P)$ in $x+P$;

$\bullet$ edge if $x$ is not flat and there is a line $L\in Gr(1,3)$ such that $x$ is an interior point of the intersection $\partial K\cap(x+L)$ in $x+L$.

Claim 1. For each flat point $x\in\partial K$ there exists a unique plane $p(x)\in Gr(2,3)$ witnessing that $x$ is flat.

Let $F$ be the set of all flat points in $\partial K$.

Claim 1 allows us to define the map $p:F\to Gr(2,3)$ assigning to each flat point $x$ the unique plane $p(x)$ witnessing that $x$ is flat.

Claim 2. The image $p(F)\subseteq Gr(2,3)$ of $F$ is at most countable.

Claim 3. For every edge point $x\in\partial K$ there exists a unique line $L\in Gr(1,3)$ witnessing thar $x$ is an edge point.

Let $E\subseteq\partial K$ be the set of edge points in $\partial K$. Claim 3 allows us to define a function $\ell:E\to Gr(1,3)$ assigning to each edge point $x$ the unique line $\ell(x)$ witnessing that $x$ is an edge point.

Endow the projective plane $Gr(1,3)$ with the metric assigning to any lines $\ell_1,\ell_2\in Gr(1,3)$ the Hausdorff distance between the doubletons $\ell_1\cap S^2$ and $\ell_2\cap S^2$.

For every $n\in\mathbb N$ let $E_n$ be the set of edge points $x\in E$ for which there exist points $a,b\in\partial K$ such that $\|a-b\|=\frac1n$, $x=\frac12(a+b)$ and $[a,b]\subseteq \partial K\cap\ell(x)$. Here $\|\cdot\|$ denotes the standard Euclidean norm of the space $\mathbb R^3$.

Claim 4. For every $n\in\mathbb N$ the restriction $\ell{\restriction}_{E_n}:E_n\to Gr(1,3)$ is a Lipschitz map.

For every edge point $x\in E$ let $x+\ell(x)^\perp$ be the plane in $\mathbb R^3$ that contains $x$ and is orthogonal to the line $\ell(x)$.

Claim 5. For every $n\in\mathbb N$ and every point $x\in E_n$ there exists a neighborhood $O_x$ of $x$ in $E$ such that for every $y\in O_x$ the line $y+\ell(y)$ has non-empty intersection with the set $E_{2n}\cap (x+\ell(x)^\perp)$.

Claims 4 and 5 imply the following

Claim 6. For every $n\in\mathbb N$ and every $x\in E_n$ there exists a neighborhood $O_x$ of $x$ in $E_n$ such that $\ell(O_x)\subseteq \ell\big(E_{2n}\cap (x+\ell(x)^\perp)\big)$.

The set $E_{2n}\cap (x+\ell(x)^\perp)$ is contained in the intersection $\partial K\cap (x+\ell(x)^\perp)$ which is a topological circle of Hausdorff dimension 1. Since the map $\ell{\restriction}_{E_{2n}}$ is Lipschitz, the image $\ell(O_x)\subseteq \ell(E_{2n}\cap(x+\ell(x)^\perp))\subseteq \ell(\partial K\cap(x+\ell(x)^\perp))$ has Hausdorff dimension $\le 1$.

Then the set $\ell(E)$ has Hausdorff dimension $\le 1$ being the union of countably many sets of Hausdorff dimension $\le 1$. Consequently, the union $\bigcup_{x\in E}\ell(E)$ has Hausdorff dimension $\le 2$ and hence is Lebesgue null in $\mathbb R^3$.

Since the set $p(F)$ is countable, the union $\bigcup_{x\in F}p(x)$ of countably many planes has Hausdorff dimension $2$ and hence is Lebesgue null in $\mathbb R^3$.

Observing that the set $B$ of $K$-boundary vectors is a subset of $\bigcup_{x\in F} p(x)\cup \bigcup_{x\in E}\ell(x)$, we see that $B$ is Lebesgue null and hence has empty interior in $\mathbb R^3$.