From Ball's cube slicing inequality to Khinchin-type inequalities for negative moments

Abstract

We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Ball's cube slicing inequality.

Sharp bounds on $p$-norms for sums of independent uniform random variables, $0 < p < 1$

“…For −1 < p < 0, the behaviour is complicated by a phase transition (similar to the case of random signs as established by Haagerup in [9]). It has recently been proved in [4] that…”

“…Since this inequality holds only in a specific range of parameters, additional arguments are needed, mainly an induction on the number of summands n (similar problems were faced in e.g. [4,24,15]). In our case, this is further complicated by the fact that the base of the induction fails for large values of p (roughly for p > 0.7).…”

“…where the last equality follows from Lemma 3 because e −t 2 /6 is the characteristic function of Z/ √ 3, Z ∼ N (0, 1) (the exchange of the order of the limit and integration in the second equality can be easily justified by truncating the integral, see, e.g., (15) in [4]). In particular, if for some p and s 0 , (3)…”

Quantitative form of Ball's Cube slicing in $\mathbb{R}^n$ and equality cases in the min-entropy power inequality

“…His paper has inspired many research in convex geometry and is still very current. We refer to [11,14,16,15,6] to quote just a few of the most recent papers in the field and refer to the reference therein for a more detailed description of the literature.…”

“…The connection between Theorem 1 and Khintchine's inequalities goes as follows: as fully derived in [6], Ball's theorem can be rephrased as…”