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I stumbled upon the following question

Distance between the boundaries of a convex set and its shrunk version

and was wondering what happens if one replaces the $\sup$ by $\inf$ and changes the problem a little. More specifically:

Let $K \subseteq \mathbb{R}^n$ be a compact, convex set and $C>1$ some constant. Moreover, assume that $B_0(\delta) \subseteq K$ (i.e. the $\delta$ ball around $0$ is contained in $K$).

Is there now any way to say something about the minimal distance between $\partial CK$ and $\partial K$? I.e., $\inf\limits_{j \in \partial CK} \inf\limits_{i \in \partial K} ||j-i|| \geq f(C,\delta)$ for some $f$? Intuitively this should be true, but I cant seem to find an answer here on stackexchange and cant come up with a satisfying proof myself either.

Any references to the above problem are appreciated!

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1 Answer 1

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For any given $i \in \partial K, j \in \partial CK$, consider the half-plane with the line through the origin and $i$ as edge, and containing $j$.

In that half-plane, let $m$ be the unique point a distance of $C\delta$ away from $0$ in the direction perpendicular to $i$. Because $CK$ is closed and $B_0(C\delta) \subseteq CK$, we must have $m \in CK$. Thus, the entire triangle formed by $Ci, m,$ and $0$ is within $CK$.

Suppose $j$ is in the interior of this triangle. Then the line from $Ci$ through $j$ would intersect the line segment from $0$ to $m$ at some point $p \ne m$. But this would mean $\|p\| < C\delta$, and thus $B_p(C\delta - \|p\|) \subseteq B_0(C\delta) \subset CK$. The cone connecting $Ci$ to $B_p(C\delta - \|p\|)$ would be an open neighborhood of $j$ entirely contained in $CK$, contradicting that $j$ is on its boundary. Thus $j$ cannot be within the interior of the triangle.

But the nearest point in the half-plane to $i$ which is not in the interior of the triangle is on the hypotenuse.

where $h = \|i\|$. By similar triangles, the distance from $i$ to the hypotenuse is $$\dfrac{(C-1)\delta}{\sqrt{1 + \left(\frac{\delta}{\|i\|}\right)^2}}$$

Since $i$ is on the boundary of $K$, it cannot be within $B_0(\delta)$, so $\|i\| \ge \delta$. So the largest the denominator can be is $\sqrt 2$.

Thus $$\inf \{\|i - j\| : i \in \partial K, j \in \partial CK\} \ge \dfrac{(C-1)\delta}{\sqrt 2}$$

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