# Distance between closed convex curve and its scaled version

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!

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