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I am stuck at the following exericse:

Let $\rho$ denote the local feature size. Draw an example where the curve C contains the line segment from $(-1, 0)$ to $(1, 0)$, but where $\rho( (0,0) )$ is arbitrarily small.

Here are the definitions that I heard in class:

Let $C$ be a smooth closed curve in the plane, and let $x$ be a point of $C$. The local feature size $\rho(x)$ of x is the shortest distance from $x$ to the medial axis of C. (The medial axis $M(C)$ of a closed smooth curve $C$ is the locus of the centers of disks that touch $C$ at two or more distinct points.)

I do not understand how it should be possible for the local feature size to become arbitrarily small, since we are looking at smooth plane curves. I have tried to draw some examples, but the ones I found were always non-smooth, like the one below. (The image comes from Wikipedia's article on cusps.)

enter image description here

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As you have suspected, it is impossible for a simple smooth closed curve to contain the line segment from (−1,0) to (1,0) such that its local feature size at $(0,0)$ is arbitrarily small.

On other hand, it is easy to draw a self-intersecting smooth curve with that property. Here is an example.

enter image description here

In fact, for every smooth curve that contains the line segment from (−1,0) to (1,0) and intersects itself at $(0,0)$, its local feature size at $(0,0)$ will be arbitrarily small.

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