Let A∈R^n× n, let e=(1,…,1)^T∈R^n be the all-ones vector, and write |A| for the entrywise absolute value of A. Suppose that |A|e=ne. Rump's 100 Euro Conjecture. There exists x∈R^n setminus0 such that |Ax|≥ |x|. The condition means that every row of A lies on the boundary of the ell_1-ball of radius n. The paper's abstract says that this conjecture is confirmed using a finite-dimensional reformulation of Ball's plank theorem, so its status is solved.

Rump's 100 Euro Conjecture for matrices

Rump posed the conjecture in 1997: every real matrix whose rows have absolute-value sum $n$ should expand some nonzero vector coordinatewise in absolute value. The retrieved paper claims this statement is true.

Known results

Rump, 1997: every admissible matrix has some nonzero $x$ satisfying $|Ax|\ge \frac{1}{3+2\sqrt{2}}|x|$ .

March 8, 2026 claimed proof

On March 8, 2026, Plank theorems, Gaussian probabilistic estimates and Rump’s 100 Euro conjecture claimed a stronger cube-escape theorem: if every row has $ell_1$ -norm at least $nt$ , then some $x\in[-1,1]^n$ satisfies $|Ax|\ge t e\ge t|x|$ . Taking $t=1$ yields the conjecture. This is a claimed proof and remains unverified.

Current status (as of September 2026): A March 2026 preprint claims to settle the conjecture via Ball’s plank theorem, but the claim remains unverified.

Rump's 100 Euro Conjecture for matrices

Primary source

Progress summary

Solutions 0