The n-Category Café

The Convex Magnitude Conjecture

Posted by Tom Leinster

For a finite subset $B = \{b_1, \ldots, b_n\}$ of $\mathbb{R}^N$, let $Z$ be the $n \times n$ matrix with $(i, j)$-entry $e^{-\left|b_i - b_j\right|}$, and define $|B|$ to be the sum of all $n^2$ entries of $Z^{-1}$.

For a compact subset $A$ of $\mathbb{R}^N$, define $|A|$ to be the supremum of $|B|$ over all finite subsets $B$ of $A$.

The 2-dimensional case of the convex magnitude conjecture states that for all compact convex $A \subseteq \mathbb{R}^2$,

$$|A| = \chi(A) + \frac{1}{4}perimeter(A) + \frac{1}{2\pi}area(A).$$

I just came back from the British Mathematical Colloquium in Sheffield, where I spoke about the convex magnitude conjecture and attempts to settle it.

Click the picture for slides.

A couple of footnotes: first, the definitions above do make sense. That is, the matrix $Z$ is always invertible, and the definition of the magnitude $|A|$ of a compact set $A$ is consistent with the definition for finite sets (which are, of course, compact too). Second, the definition of $|A|$ for compact $A$ isn’t the same as the one given in the slides, but it’s equivalent. Clickable references for everything are in the final slide.