**
A robust Khintchine inequality, and
algorithms for computing optimal constants in Fourier analysis and
high-dimensional geometry.
**

A. De and
I. Diakonikolas and
R. Servedio.

*40th International Colloquium on Automata, Languages and Programming
(ICALP)*, 2013, pp. 376--387.

**Abstract:**
This paper makes two contributions towards determining some well-studied
optimal constants in Fourier analysis of Boolean functions
and high-dimensional geometry.

It has been known since 1994 \cite{GL:94}
that every linear threshold function has squared Fourier mass
at least $1/2$ on its degree-$0$ and degree-$1$ coefficients.
Denote the minimum such Fourier mass by
$\w^{\leq 1}[\ltf]$, where the minimum is taken over all $n$-variable
linear threshold functions and all $n \ge 0$.
Benjamini, Kalai and Schramm \cite{BKS:99}
have conjectured that the true value of $\w^{\leq 1}[\ltf]$ is $2/\pi$.
We make progress on this conjecture by proving that $\w^{\leq 1}[\ltf]
\geq 1/2 + c$ for some absolute constant $c>0$.
The key ingredient in our proof is a ``robust'' version of the well-known
Khintchine inequality in functional analysis, which we
believe may be of independent interest.

We give an algorithm with the following property: given any $\eta > 0$,
the algorithm runs in time $2^{\poly(1/\eta)}$ and determines the value of
$\w^{\leq 1}[\ltf]$ up to an additive error of $\pm\eta$. We give a similar
$2^{{\poly(1/\eta)}}$-time
algorithm to determine *Tomaszewski's constant*
to within an additive error of $\pm \eta$; this is the
minimum (over all origin-centered hyperplanes $H$) fraction of points
in $\{-1,1\}^n$ that lie within Euclidean distance $1$ of $H$.
Tomaszewski's constant is conjectured to be $1/2$; lower bounds on it
have been given by Holzman and Kleitman \cite{HK92} and
independently by Ben-Tal, Nemirovski and Roos
\cite{BNR02}.
Our algorithms combine tools from anti-concentration
of sums of independent random variables, Fourier analysis, and Hermite
analysis of linear threshold functions.

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