Convex Influences

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Defines convex influence along a direction as the negated degree-two Hermite coefficient and shows it equals the infinitesimal rate at which Gaussian volume changes under directional dilation. Proves influences are nonnegative, zero influence implies the body is a cylinder (measure‑zero exception), and total influence is rotationally invariant. Obtains analogs of Poincaré and a KKL-type bound: if each direction has influence ≤ ε then total influence ≥ Var(K)·sqrt(log(Var(K)/ε)), using Gaussian isoperimetry and Brascamp‑Lieb (Vempala) methods, not hypercontractivity.

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" chapters (19)

1. 0:00An Intriguing Analogy
2. 2:02Monotone Boolean Functions & Symmetric Convex Sets
3. 4:08Towards Quantitative Inequalities
4. 5:22Influence of Variables on Boolean Functions
5. 6:40Talagrand's Correlation Inequality (1996)
6. 7:07Story So Far
7. 10:14Convex Influence in Terms of Dilations
8. 12:43Total Convex Influence
9. 14:08A Sanity Check
10. 14:38Are Convex Influences Non-Negative?
11. 15:39Convex Influences Are Non-Negative
12. 17:03Zero Convex Influences Implies Cylinder
13. 18:17Boolean Dictatorships are like Slabs
14. 19:16Majority is like the Ball
15. 20:12Tribes is like the Solid Cube
16. 22:41The KKL Theorem
17. 23:31KKL for Symmetric Convex Sets
18. 24:51Proof Sketch
19. 26:09Several Open Directions