Dembo–Cover–Thomas conjecture and related projection inequalities

The Dembo–Cover–Thomas conjecture concerns volume inequalities for Minkowski sums and equivalent projection inequalities. Recent work claims to refute its zonoid projection formulation in dimensions $n\ge 3$, without resolving every conjecture-motivated variant.

Known results

• The inequality holds for arbitrary convex bodies in dimension $2$ but is false in general for every $n\ge 3$; it holds for zonoids in dimension $3$ in a restricted form (2022).
• A 2026 manuscript gives an explicit three-dimensional zonoid counterexample and extends it to every $n>3$ by taking products with a cube.
• The same manuscript proves certain degree or codegree at most $3$ cases and reports failure of a weaker related conjecture for zonoids in every $n\ge 4$.

2026 determinant and projection counterexamples

A manuscript titled Volume and Projection Inequalities II: Determinants and $L_p$-Sums claims that the stated strong inequality fails in dimensions $n\ge 2$ for $1<p<2$, proves a planar weak inequality, and establishes dimension-dependent validity and failure ranges for determinant-power analogues. These claims are unverified, and the abstract does not assert that every formulation of the Dembo–Cover–Thomas conjecture is resolved.

Current status (as of August 2026): The principal zonoid projection formulation is claimed false for $n\ge 3$, with the two-dimensional case known to hold, while related formulations and the latest $L_p$ claims remain unverified or unresolved.