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March 30, 2013

The Convex Magnitude Conjecture

Posted by Tom Leinster

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

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

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

|A|=χ(A)+14perimeter(A)+12πarea(A). |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.

Title slide of talk

Click the picture for slides.

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

Posted at March 30, 2013 5:48 PM UTC

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Re: The Convex Magnitude Conjecture

Incidentally, several people remarked that the photo of Simon — which is a still from a video recorded some years ago — doesn’t really look like him. But then it was observed that the shadow Simon casts on the projector screen is somehow a much better resemblance. Simon’s reaction? “I’m like a shadow of my former self.”

Posted by: Tom Leinster on March 30, 2013 6:40 PM | Permalink | Reply to this
Read the post Magnitude of Metric Spaces: A Roundup
Weblog: The n-Category Café
Excerpt: Resources on magnitude of metric spaces.
Tracked: April 7, 2013 9:57 PM

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