## Barnabás Janzer: Rotation inside convex Kakeya sets

Barnabás Janzer studied the following question:

Suppose we have convex body $K$ in $\mathbb R^n$ that contains a copy of a convex body $S$ in every orientation. Is it always possible to move any one copy of $S$ to another copy of $S$, keeping inside $K$?

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 $\mathbb R^d$ has Housdorff dimension $d$. 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 $\mathbb Z$ or $\mathbb R$ 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 $K$ in $\mathbb R^n$ that contains a copy of a convex body $S$ in every orientation. Is it always possible to move any one copy of $S$ to another copy of $S$, keeping inside $K$?

Barnabás Janzer proved:

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

Here is the link to the paper:

### Rotation inside convex Kakeya sets

This entry was posted in Convexity, Test your intuition and tagged , . Bookmark the permalink.

### 1 Response to Barnabás Janzer: Rotation inside convex Kakeya sets

1. domotorp says:

And here is a link to his online talk next Friday: https://coge.elte.hu/seminar.html