Multivariable Calculus Challenge Problem

[crimsix]

Homework Statement

Here it is:

Let Ω be a convex region in R2 and let L be a line segment of length ι that connects points on the boundary of Ω. As we move one end of L around the boundary, the other end will also move about this boundary, and the midpoint of L will trace out a curve within Ω that bounds a (smaller) region Γ. Find an expression that relates the area of Γ to the area of Ω in terms of the length ι.

Homework Equations

See above.

The Attempt at a Solution

I have no idea. This is well beyond the scope of our course. The instructor put this out to get his students to reach out to the mathematics community and get involved in discussions of (multivariable) calculus. The closest I have come to encountering such a problem is related rates, but I have no idea how to do this.

 

[Ray Vickson]

Homework Statement

Here it is:

Let Ω be a convex region in R2 and let L be a line segment of length ι that connects points on the boundary of Ω. As we move one end of L around the boundary, the other end will also move about this boundary, and the midpoint of L will trace out a curve within Ω that bounds a (smaller) region Γ. Find an expression that relates the area of Γ to the area of Ω in terms of the length ι.

Homework Equations

See above.

The Attempt at a Solution

I have no idea. This is well beyond the scope of our course. The instructor put this out to get his students to reach out to the mathematics community and get involved in discussions of (multivariable) calculus. The closest I have come to encountering such a problem is related rates, but I have no idea how to do this.

While you may not be able to do it in general (it seems really hard!) you might have some luck with special regions such as circular or elliptical disks, or maybe rectangles. Why not try the (seemingly) easier cases first? The concept of "envelope of lines or curves" seems to be relevant.