# How can the local feature size become arbitrarily small?

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.)

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.
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.