Displacement-Energy Bound for Plank Coverings

Definitions and mathematical statement

Let $n\geq 1$, and let $K\subset{\mathbb{R}}^n$ be a convex body, meaning a compact convex set with nonempty interior. A plank is a set \[ P_i=\{x\in{\mathbb{R}}^n:a_i\leq\ell_i(x)\leq b_i\}, \qquad 0\neq\ell_i\in({\mathbb{R}}^n)^*,\quad a_i<b_i. \] Consider a finite covering $K\subseteq\bigcup_{i=1}^{m}P_i$, where $m\geq1$. Let $B$ be the unit ball of the norm mentioned in the supplied statement. Its polar is \[ B^\circ=\{p\in({\mathbb{R}}^n)^*:|p(x)|\leq1\text{ for every }x\in B\}. \] Thus $K\times B^\circ$ lies in ${\mathbb{R}}^n\times({\mathbb{R}}^n)^*$.

For clarity, the standard symplectic interpretation uses coordinates $(x,p)$ and the form $\omega_0=\sum_{j=1}^{n}{\,\mathrm{d}} x_j\wedge{\,\mathrm{d}} p_j$. Under the usual Hofer normalization, the displacement energy of a compact set $A$ in this space is \[ e(A)=\inf_H\int_0^1\left(\max_z H(t,z)-\min_z H(t,z)\right){\,\mathrm{d}} t, \] where the infimum is over smooth, compactly supported Hamiltonians whose time-one Hamiltonian flow satisfies $\varphi_H^1(A)\cap A=\varnothing$. An empty infimum is interpreted as $+\infty$. These are conventional explanatory definitions; the supplied record does not explicitly specify the ambient symplectic structure or the energy normalization.

Write $r_i$ for the relative width assigned to $P_i$ in the intended source convention, and put $w=\sum_{i=1}^{m}r_i$. The supplied conjectural question is whether every such covering satisfies \[ e(K\times B^\circ)\leq2w. \] This is a statement-only reconstruction with an unresolved normalization. In particular, if \[ h_B(\ell)=\sup_{x\in B}\ell(x), \] then the width measured as distance between the boundary hyperplanes in the norm with unit ball $B$ is $(b_i-a_i)/h_B(\ell_i)$, whereas the ratio of the plank's width to the width of $B$ in the same direction is $(b_i-a_i)/(2h_B(\ell_i))$. Relative width with respect to $K$ would instead use the denominator $\sup_{x\in K}\ell_i(x)-\inf_{x\in K}\ell_i(x)$. The supplied record does not select a formula for $r_i$, specify any required relation between $K$ and $B$, or state the intended dimensional range. These omissions prevent a unique fully quantified formulation with the numerical constant $2$; no width convention or additional relation is asserted here.

Short English statement

The supplied problem asks whether a finite plank covering of a convex body bounds the displacement energy of its product with a polar norm ball by twice the sum of the planks' relative widths. The meaning of relative width, the relation between the convex body and the norm ball, and the symplectic normalization require clarification before the constant can be interpreted precisely. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30002662$, title "Displacement-Energy Bound for Plank Coverings," source problem code catalogue entry. This draft uses the complete registry record supplied in the prompt; it does not claim independent inspection of the registry file.
• Supplied source citation: Discrete Geometry (2015), Oberwolfach Reports, DOI 10.4171/owr/2014/40, pp. 2235--2296. Original report link: https://doi.org/10.4171/owr/2014/40. The record describes the collection as arising from workshop open-problem discussions. No individual proposer is supplied.
• Supplied literature source: Akopyan, Karasev, and Petrov, "Bang problem," https://arxiv.org/abs/1404.0871. No more precise bibliographic attribution is supplied.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and states that the displacement-energy plank inequality is proved only in special configurations, while the full conjecture remains open together with the general non-symmetric Bang problem. This is supplied dated curation, not independently verified here; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach passage was checked directly and its question, definitions, and notation restored without strengthening the claim. These are supplied provenance assertions, not verification performed for this draft. The record gives proposed year $2014$, while its report citation says $2015$ and its DOI contains $2014$; these details are retained without reconciliation.