# On convex bodies

Dmitry Ryabogin from Kent State University gave two talks today here at UCalgray. The first talk was about convex bodies with congruent projections. You can read about it here: PDF. And their recent results here: arXiv. The question asks for two convex bodies $k$ and $L$ in $\mathbb{R}^n$, if orthogonal projections of them into each hyperplane is a rotation and translation of each other, is it true that $K = \pm L + a$ where $a$ is a translation?

One side interesting thing is the following question: Does there exist a convex body $C$ in $\mathbb{R}^2 (\mathbb{R}^n)$, which is not a disk (sphere), such that it can be rotated arbitrarily between two parallel lines (hyperplanes) without loosing its contact with either one of them.

It turns out that such things are well known and they are called convex bodies of constant width. It goes back at least to Euler. Here is one way to construct one in $\mathbb{R}^2$: Start with an equilateral triangle ABC and draw  a circle centred at A and with radius AB=AC. Repeat this for other two vertices. The intersection of the circles can be shown that has constant width. The more interesting part is that if you use this as a drill you’ll make a whole that is a square! Guess what’s the length of the square!

This doesn’t quite make a wheel in the conventional way that the axle is connected to the centre. But if you put something on top of 4 of them, it’ll always remain parallel to the ground, of course. Here is a link from Wikipedia. And you can read more about them in Chapter 3 (Convex Bodies of Constant Width) of the book Convexity and Its Applications, by Chakerian and Groemer.

From Wikipedia: Rouleaux triangle

A naive question is if we consider all projections into all subspaces, rather than just hyperplanes, does this question become trivial, or is it equivalent to the original problem? Or maybe something else?