Khintchine Polytope Reduction

• Khintchine polytope reduction is a method that reduces the optimal constants in multilinear inequalities to functions of the single-variable case using tensorization.
• It employs geometric analysis and techniques like Minkowski’s inequality to establish sharp, dimensionally extended bounds in Rademacher series.
• The approach unifies constants in multiple Khintchine and mixed Littlewood inequalities, simplifying high-dimensional analysis with practical tensor product structures.

The Khintchine polytope reduction refers to the structural phenomenon whereby the best constants in inequalities involving multiple Rademacher variables—specifically, the multiple Khintchine inequality—can be systematically and explicitly reduced to functions of the optimal constant in the classical, single-variable Khintchine inequality. This reduction elucidates the geometric and analytic relationship between classical and multilinear inequalities, and establishes a clear blueprint for deriving optimal bounds in higher-dimensional scenarios.

1. The Multiple Khintchine Inequality and Its Constants

The multiple Khintchine inequality is an extension of the classical Khintchine inequality to arrays of coefficients indexed by $m$ variables. For $0 < r < \infty,$ $m \in \mathbb{N},$ and scalars $(y_{i_1, \ldots, i_m})_{i_1, \ldots, i_m=1}^N$, there exists a constant $K_{m,r} \geq 1$ such that

$$\left(\sum_{i_1, \ldots, i_m = 1}^{N} |y_{i_1, \ldots, i_m}|^2\right)^{1/2} \leq K_{m,r} \left( \int_{[0,1]^m} \left| \sum_{i_1, \ldots, i_m} r_{i_1}(t_1) \cdots r_{i_m}(t_m) y_{i_1, \ldots, i_m} \right|^r dt_1 \cdots dt_m \right)^{1/r},$$

where $r_j$ are the Rademacher functions. The best constant is given by

$K_{m,r} = (A_r)^m,$

where $A_r$ is the optimal constant in the classical Khintchine inequality: $$\left( \int_0^1 \left| \sum_{j=1}^n a_j r_j(t) \right|^r dt \right)^{1/r} \leq A_r \left( \sum_{j=1}^n |a_j|^2 \right)^{1/2}.$$

Explicitly, $A_r$ is: $$A_r = \begin{cases} 2^{1/2 - 1/r}, & 0 < r \leq p_0, \ \sqrt{2}\left(\dfrac{\Gamma\left( \frac{r+1}{2} \right)}{\sqrt{\pi}}\right)^{1/r}, & p_0 < r < 2, \ 1, & r \geq 2, \end{cases}$$ where $p_0$ is the unique solution of $$\Gamma(p_0+1) = \sqrt{\pi}\,\Gamma\left(\frac{p_0}{2} + 1\right)$$, and $\Gamma$ is the gamma function. This provides sharp constants for all $m$ and all values of $r$.

2. Geometric Structure: The Khintchine Polytope

The Khintchine polytope refers to the set of all possible vectors of $L^p$-moments, $(\int|\sum_j a_j r_j|^p dt)^{1/p}$, for sequences with a fixed $\ell_2$-norm. In the multiple case, the corresponding polytope comprises all $m$-tuple moment vectors for arrays satisfying the multiple Khintchine inequalities. The geometric structure of the multiple Khintchine polytope is a Cartesian product of the structure in the single-variable case. Thus, extremal or equality cases in the $m$-variable setting are tensor products of extremal cases in the classical setting.

3. Reduction Principle and Product Structure

The reduction principle asserts that because of the independence of the Rademacher functions across coordinates, the best constant in the multilinear, $m$-variable case is simply the $m$-th power of the best constant in the classical (single-variable) case. If $A_r$ describes the "radius" or the maximal norm-preserving distortion in the one-dimensional polytope, then the multiple case's "radius" is $(A_r)^m$. Consequently, optimal arrays in the $m$-variable inequality are tensor products of optimal vectors from the classical scenario.

This product structure is both algebraic (constants) and geometric (shape of the polytope), reflecting the independence and tensorization properties in the analysis.

4. Analytical and Proof Techniques

The sharp constants are rigorously determined via induction on $m$, application of Minkowski’s integral inequality, Fubini’s theorem, and careful analysis of extremal arrays (notably those with $(0,1)$ coefficients or all ones). Central limit arguments and results on the best constant in the classical inequality (notably Haagerup’s solution) are essential. The findings are consistent with what extremal cases suggest, and the passage to the limit as $N \to \infty$ is critical for identifying sharpness.

5. Consequences for Mixed Littlewood Inequalities

A key implication is the unification of optimal constants for the multiple Khintchine and the multilinear mixed $(\ell_p,\ell_2)$-Littlewood inequalities. The latter's best constant is also a power of $A_p$: $\text{Best constant for %%%%28%%%%-linear mixed Littlewood:} \quad (A_p)^{m-1}.$ Therefore, these classes of inequalities are equivalent with respect to their optimal constants.

6. Table of Optimal Constants

Inequality Type	Best Constant	Explicit Expression
Classical Khintchine	$A_r$	See formulas above
Multiple Khintchine ($m$-variable)	$K_{m,r}$	$(A_r)^m$
Mixed $(\ell_p,\ell_2)$-Littlewood	$(A_p)^{m-1}$	$(A_p)^{m-1}$

7. Implications and Structural Simplification

The polytope reduction underlines a substantial structural simplification: for all $m$, analysis of the multiple Khintchine inequality reduces to understanding the classical case. This equivalence highlights the profound influence of independence and product structure in the theory of inequalities for Rademacher series. The results finalize previous lines of inquiry, correcting and completing the determination of all optimal constants for both the multiple Khintchine and the mixed Littlewood settings. All extremal examples and sharpness constructions are inherited directly via tensorization from known cases for the classical (single) Khintchine polytope.