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Inspired by Steinitz's celebrated result (1913) and recent online vector-balancing results of Kulkarni et al. (2024) and Aden-Ali (2026), we introduce the online Steinitz problem: $T$ vectors in $\mathbb R^d$, each of $\ell_2$-norm at most $1$ and with total sum zero, arrive one at a time and must be assigned immediate...
We consider $0/1$ packing problems $\max\{c^T x \colon Ax \leq 1, \, x \in \{0,1\}^n\}$, with $A \in \mathbb{R}_{\geq 0}^{m \times n}$. A way to solve such problems is via tightening the linear programming relaxation $P$ with Gomory \emph{cutting-planes}. The Gomory-closure $P'$ of $P$ is the intersection of $P$ with a...
We present a short proof of Feige's conjecture: for $n$ independent nonnegative random variables with expectation one, the probability that their sum is less than $n+1$ is at least $\left(\frac{n}{n+1}\right)^n\ge \frac{1}{e}$. The proof was obtained with the assistance of GPT-5.6 Sol and builds on the recent breakthro...
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