Nearly-Tight Bounds for Zonotope Containment and Beyond

2026-05-05

Abstract

We investigate the convex-body containment problem $\max\{s >0 : s Z \subseteq Q\}$, where the outer body $Q \subseteq \mathbb R^d$ is described by a membership oracle and the inner body $Z \subseteq \mathbb R^d$ is a zonotope. Our main result is a sampling-based $O(\sqrt{d})$-approximation algorithm for this problem that almost matches the lower bound of $Ω(\sqrt{d/\log d})$ by Khot and Naor in the oracle model. Assuming zonotopes can be sparsified by a linear number of generators, which is referred to as Talagrand conjecture, our approach attains the optimal approximation factor of $Θ(\sqrt{d/\log d})$. Our second main result is a proof of Talagrand's conjecture for $Δ$-modular zonotopes whenever $Δ$ is constant. Those zonotopes are of the form $Z = \{ Wx \colon \| x\|_\infty \leq 1\}$ where the non-zero $d \times d$ sub-determinants of $W$ are between $1$ and $Δ$. This result establishes a connection between zonoid sparsification and spectral sparsification of Batson, Spielman and Srivastava. We complement these results with a universal $Ω(\sqrt{d/\log d})$ lower bound holding for all zonotopes. Finally, we consider containment problems $\max\{s >0 : s K \subseteq Q\}$, for general convex bodies $K \subseteq \mathbb R^d$. A result of Naszódi on approximating $K \subseteq \mathbb R^d$ by a polytope implies a $Θ(d/\log d)$ approximation algorithm in polynomial time. We show the tightness of this approximation factor in the oracle model via a reduction to the circumradius computation. Our lower bound holds for centrally symmetric convex sets, implying that Barvinok's optimal $O(\sqrt{d})$-approximation of a centrally symmetric convex body by a polytope with a polynomial number of vertices cannot be computed in polynomial time.

Problem

• convex-body containment
• zonotope
• approximation algorithm
• oracle model
• Talagrand conjecture

Innovation

• zonotope containment
• approximation algorithms
• Talagrand conjecture
• convex body approximation
• oracle complexity