Archive for April 2010

Infinitely Many Primes by Combinatorics

Here’s a proof of the infinitude of primes that occurred to me when for some reason Delta upgraded me to First Class on a flight April 2. Can anyone provide a reference?

Suppose that the set P of primes and 1 has just n+1 elements. Now every number at most 2k can be obtained by choosing k elements of P with replacement, which can be done in (k+n choose n) ways. Therefore

2k ≤ (k+n choose n) ≤ (k+n)n ,

which fails for k large.

The Log-Convex Density Conjecture

Announcement: proof posted 19 November 2013 by Gregory R. Chambers, student of Regina Rotman and Alex Nabutovsky at Toronto. By spherical symmetrization the problem reduces to planar curves, which he studies very intelligently in great detail. The main idea is that if the generating curve is not a circle about the origin, then from its maximum (on the axis of symmetry) it spirals inward and it eventually turns through 2π before returning to the axis, contradiction. See his 30-second video. “Proof of the Log-Convex Density Conjecture,” J. Eur. Math. Soc. 21 (2019), 2301–2332.

Update August 2, 2022. arXiv:2208.00195. “Approaching the isoperimetric problem in H^m_ℂ via the hyperbolic log-convex density conjecture” by Lauro Silini. Abstract: We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space H^n_ℝ endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on ℝ^n. As an application we prove that in the complex and quaterionic hyperbolic spaces, geodesic balls are isoperimetric in a class of sets enjoying a suitable notion of radial symmetry. The hyperbolic plane case was proved by Igor McGillivray: arXiv:1712.07690, “A weighted isoperimetric inequality on the hyperbolic plane.”

Ken Brakke’s Log-Convex Density Conjecture [Rosales et al. Conj. 3.12] says that in Euclidean space with radial log-convex density f(r), balls about the origin are isoperimetric.

A density is just a positive continuous function used to weight volume and perimeter. Log convexity just means that log f is convex. Balls isoperimetric means that any other region of the same weighted volume has no less weighted perimeter.

Log convexity is necessary because it is equivalent to stability. The question is whether it is sufficient. Continue reading ‘The Log-Convex Density Conjecture’ »