Barnabás Janzer: Rotation inside convex Kakeya sets

Barnabás Janzer studied the following question:

Suppose we have convex body in that contains a copy of a convex body in every orientation. Is it always possible to move any one copy of to another copy of , keeping inside ?

I will let you test your intuition about what the answer should be. First, some background.

A Kakeya set is a set that contains unit unterval in every direction. A famous open problem is the conjecture that every Kakeya set in has Housdorff dimension . in 2008, Zeev Dvir found a simple remarkable proof for a finite field analog of the conjecture. Finding possible connections between the finite field problem and the Euclidean problem is an exciting problem. Can we use the finite field result to prove the Euclidean result? Can we use or refine the finite field methods for the Euclidean problem? Here is a recent Quanta Magazine article about exciting “intermediate results”.

The question about the connection between finite fields analogs and questions over or can be asked about other problems. One example is the Roth problem (relevant posts I,II, III) vs. the cup set problems (relevant posts I,II,III).

Going back to our problem (posed by H. T. Croft) about convex Kakeya sets:

Suppose we have convex body in that contains a copy of a convex body in every orientation. Is it always possible to move any one copy of to another copy of , keeping inside ?

Barnabás Janzer proved:

The answer is yes in 2 dimensions:

The answer is also yes in dimensions if is one-dimensional (i.e. an interval).

But, amazingly: the answer is no in general in 4 dimensions!

Here is the link to the paper:

Rotation inside convex Kakeya sets