Lower Estimates of Marginal Density
summary
Establishes a concrete lower bound for the marginal density of the uniform cube that depends only on the projected dimension d and not on ambient n, with a proof strategy combining Fourier/characteristic function reductions, a compressible/incompressible decomposition of directions, and log concavity stability to turn probability lower bounds into density lower bounds. For one dimensional hyperplane sections the argument gives a usable numeric bound about 1/17 by rewriting volume as a positive S2 integral and controlling a quadratic form via Laplace transform and eigenstructure, losing a factor ~5.2 from optimal.
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chapters (14)
- 0:00Start
- 1:30Marginal density bounds
- 6:11Sections of a cube
- 9:56Lower bound - general dimension
- 11:00Proof ideas
- 12:14Step 1: vectors with a large la norm.
- 26:35A "reasonable" bound
- 27:48Fourier analysis at work
- 33:06From the integral to probability
- 35:07Probability of positivity
- 39:09Exponential moment
- 40:34Laplace transform
- 44:44Probability vs geometry
- 47:04Step 3: compressible vectors.