A support-discrepancy formulation for zonotope generator partitions

Definitions and mathematical statement

The original question, attributed to Leonid Gurvits, asks for an analogue of Kadison--Singer for $\mathrm{Zon}=\sum_{i=1}^k\alpha_i[0,z_i]$, with $z_i\in{\mathbb{R}}^n$, $k>n$, but states neither coefficient conditions nor a desired conclusion. The aid adopts a centered, consistently weighted repair: fix vectors $v_1,\ldots,v_k\in{\mathbb{R}}^n$, not all zero, put $E=\operatorname{span}\{v_i\}$, $d=\dim E$, and \[ Z=\sum_{i=1}^k[-v_i,v_i],\qquad Z_J=\sum_{i\in J}[-v_i,v_i],\qquad h_Z(x)=\sum_{i=1}^k|\langle v_i,x\rangle|\quad(x\in E). \] All weights are absorbed into the $v_i$, and the same weighted generators are used in both subzonotopes. The relation of these repaired generators to the unspecified original coefficients is not uniquely determined.

For $0\leq\eta\leq1/2$, simultaneous approximate-half inclusions mean \[ (1/2-\eta)Z\subset Z_J\subset(1/2+\eta)Z,\qquad (1/2-\eta)Z\subset Z_{J^c}\subset(1/2+\eta)Z. \] Writing $\varepsilon_i=1$ on $J$ and $-1$ on $J^c$, these are equivalent to \[ \left|\sum_{i=1}^k\varepsilon_i|\langle v_i,x\rangle|\right| \leq2\eta\sum_{i=1}^k|\langle v_i,x\rangle|\qquad(x\in E). \] Define the actual small-segment parameter, rather than an arbitrary upper bound for it, by \[ \delta=\max_i\sup_{x\in E\setminus\{0\}} \frac{|\langle v_i,x\rangle|}{h_Z(x)} =\min\{a\geq0:[-v_i,v_i]\subset aZ\text{ for every }i\}. \] Then $0<\delta\leq1$. The reported partial theorem provides a partition with relative support discrepancy \[ \sup_{x\in E\setminus\{0\}} \frac{|\sum_i\varepsilon_i|\langle v_i,x\rangle||}{h_Z(x)} \leq\min\{1,12\sqrt{3\delta d}\}, \] hence with $\eta\leq\min\{1/2,6\sqrt{3\delta d}\}$. The bound is also valid with any larger prescribed small-segment upper bound in place of $\delta$.

In the opposite direction, every partition has relative discrepancy at least $\max\{0,2\delta-1\}$, where $\delta$ must be the actual parameter just defined. This lower bound does not follow from an arbitrarily loose inclusion parameter. The aid reports a full-dimensional volume-one box with $\delta=1$, showing that volume normalization alone cannot force a nontrivial approximate-half partition; zero generators may be appended if necessary to meet the original $k>n$ convention.

For the parallel-class refinement, a precise within-class ratio can be defined as follows: divide the nonzero generators into parallel classes $C_\ell$, write $v_i=c_iw_\ell$ in each class, and put \[ \rho=\max_\ell\frac{\max_{i\in C_\ell}|c_i|}{\sum_{i\in C_\ell}|c_i|}. \] With this explicitly stated convention, balancing each class yields the reported error form $\eta\leq\rho/2$. The summary does not itself define its symbol $\rho$, so this normalization is identified as part of the precise repaired formulation. None of these dimension-dependent partial bounds recovers a uniquely specified original zonotope analogue of Kadison--Singer.

Short English statement

The source asks an incomplete question about a zonotope analogue of Kadison--Singer. The supplied repair asks to split the same centered generators into two approximate halves, exactly equivalent to a uniform support-function discrepancy bound. It gives a dimension-dependent small-segment estimate and shows that volume normalization alone cannot replace directional smallness.

Sources

• Local registry: permanent UnsolvedMath ID $20000567$, exact public code catalogue entry; aim-analysis-notes.json, notes[98].
• AIM workshop Beyond Kadison--Singer: paving and consequences, item b26, attributed in the supplied verification narrative to Leonid Gurvits: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit.
• Supplied research summary, "A support-discrepancy formulation for zonotope generator partitions," attempt 1, reports a multiscale Rademacher-net proof, lower obstruction, volume-one example, and parallel-class refinement. No supporting external theorem citation is supplied; original remarks and literature fields are empty.