Affine plank problem¶
The conjecture that planks covering a convex body have total relative width at least one.
Core Idea¶
Relative width divides each plank width by the body’s width in the perpendicular direction, making the statement affine invariant; the general case remains open. Each covering strip is normalized against the same body in its own direction, and the conjecture lower-bounds the sum required to cover every point. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
The load-bearing residual is not the broad topic of convex geometry. It is the domain-specific identity fixed by the dimension and convex body, covering planks and normals, ordinary and body widths, relative-width formula, coverage condition, lower-bound claim, proven special cases and open status are explicit.
Scope of Application¶
Affine plank problem belongs to convex geometry and is useful where the analyst can specify the typed convex geometry carrier, including its objects, relations, parameters, conventions, evidence, and comparison cases, then evaluate the dimension and convex body, covering planks and normals, ordinary and body widths, relative-width formula, coverage condition, lower-bound claim, proven special cases and open status are explicit. The scope is broad within that domain but bounded by the need for the dimension and convex body, covering planks and normals, ordinary and body widths, relative-width formula, coverage condition, lower-bound claim, proven special cases and open status are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the dimension and convex body, covering planks and normals, ordinary and body widths, relative-width formula, coverage condition, lower-bound claim, proven special cases and open status are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Affine plank problem. Affine plank problem compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed convex geometry carrier, including its objects, relations, parameters, conventions, evidence, and comparison cases. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the dimension and convex body, covering planks and normals, ordinary and body widths, relative-width formula, coverage condition, lower-bound claim, proven special cases and open status are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of convex geometry because they reuse the typed convex geometry carrier, including its objects, relations, parameters, conventions, evidence, and comparison cases, Each covering strip is normalized against the same body in its own direction, and the conjecture lower-bounds the sum required to cover every point., and type the carrier, state every parameter and convention in the definition, test that the dimension and convex body, covering planks and normals, ordinary and body widths, relative-width formula, coverage condition, lower-bound claim, proven special cases and open status are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Affine plank problem Domain-specific
Parents (1) — more general patterns this builds on
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Affine plank problem is a kind of Coverage / Reachability Prime
The proposed strict upward parent is
prime:coverage_reachability.
Hierarchy paths (2) — routes to 2 parentless roots
- Affine plank problem → Coverage / Reachability → Completeness
- Affine plank problem → Coverage / Reachability → Surjectivity → Function (Mapping)
Neighborhood in Abstraction Space¶
Affine plank problem sits in a crowded region of the domain-specific corpus (24th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Convex Geometry & Spatial Partition (35 abstractions)
Nearest neighbors
- Convex hull — 0.94
- Supporting hyperplane — 0.93
- Equichordal point problem — 0.91
- Relative convex hull — 0.91
- Binary space partitioning — 0.90
Computed from structural-signature embeddings · 2026-09-08