Isoperimetry for spherically symmetric log-concave probability measures

Small ball probability estimates for log-concave measures

We establish a small ball probability inequality for isotropic log-concave probability measures: there exist absolute constants c1, c2 > 0 such that if X is an isotropic log-concave random vector in R with ψ2 constant bounded by b and if A is a non-zero n × n matrix, then for every ε ∈ (0, c1) and y ∈ R, P (‖Ax− y‖2 6 ε‖A‖HS) 6 ε ` c2 b ‖A‖HS ‖A‖op ́2 , where c1, c2 > 0 are absolute constants.

Functional Inequalities for Gaussian and Log-Concave Probability Measures

We give three proofs of a functional inequality for the standard Gaussian measure originally due to William Beckner. The first uses the central limit theorem and a tensorial property of the inequality. The second uses the Ornstein-Uhlenbeck semigroup, and the third uses the heat semigroup. These latter two proofs yield a more general inequality than the one Beckner originally proved. We then ge...

Isoperimetry and stability of hyperplanes for product probability measures

Classical position probability densities for spherically symmetric potentials

A simple position probability density formulation is presented for the motion of a particle in a spherically symmetric potential. The approach provides an alternative to Newtonian methods for presentation in an elementary course, and requires only elementary algebra and one tabulated integral. The method is applied to compute the distributions for the Kepler-Coulomb and isotropic harmonic oscil...

Modified Log-sobolev Inequalities and Isoperimetry

We find sufficient conditions for a probability measure μ to satisfy an inequality of the type ∫ Rd fF ( f ∫ Rd f 2 dμ ) dμ ≤ C ∫ Rd f2c∗ ( |∇f | |f | ) dμ + B ∫ Rd f dμ, where F is concave and c (a cost function) is convex. We show that under broad assumptions on c and F the above inequality holds if for some δ > 0 and ε > 0 one has ∫ ε 0 Φ ( δc [ tF ( 1t ) Iμ(t) ]) dt <∞, where Iμ is the isop...