## Log-Sobolev Inequality

My 2009 Williams College NSF “SMALL” undergraduate research Geometry Group has the following inequality for any function on the unit interval and for any p ≥ 1:

with equality for constant functions and if p>1 only for constant functions. They conjecture that these results still hold if on the right-hand side is replaced by (sharp).

The case p=1 is standard and follows from Wirtinger’s Inequality.

Are any inequalities like this known?

## LJS:

Isn’t the LHS decreasing in p, so p=1 is worst case?

This does not of course help with the conjecture.

11 June 2009, 2:36 pmRight you are, thanks very much. The undergraduates came upon it in a completely different way, as a generalization of the isoperimetric inequality (p=1) to sectors of planes with density , using the four vertex theorem. Your comment inspired a simpler geometric comparison argument. Do you believe the conjecture?—FM## Frank Morgan:

Comment from David Thompson, SMALL undergraduate research Geometry Group ’09

The conjectured inequality says that for all ,

.

The case holds trivially; is the standard isoperimetric inequality. Both sides are concave as functions of . Would some kind of interpolation argument prove the result for all ?

14 June 2009, 10:04 am## Frank Morgan:

Francesco Maggi at Firenze notes that differentiating with respect to

qatq=1 implies the following log-Sobolev inequality for uniform measure on the interval for a nonnegative functionfof mean 1:Corollary of Conjecture: .

Similarly the corollary, on small perturbations of

9 June 2010, 3:29 amf= 1, implies the sharp, known Wirtinger inequality:for (nonperiodic)

uof mean 0. (Foruperiodic the sharp constant is .)## Frank Morgan:

My 2012 undergraduate research Geometry Group has observed that the sharp log-Sobolev inequality on the interval follows immediately from a similar inequality on the circle.

30 August 2012, 5:17 am## Frank Morgan » Blog Archive » Manifolds with Density: Fuller References:

[…] [2012] Alexander Díaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; arXiv.org (2010); see blog posts 1 and 2. […]

17 October 2014, 9:16 am