90:
It is not completely trivial that a minimum-area cover exists. An alternative possibility would be that there is some minimal area that can be approached but not actually attained. However, there does exist a smallest
82:
with vertex angles of 60° and 120° and with a long diagonal of unit length. However, these are not optimal solutions; other shapes are known that solve the problem with smaller areas.
181:
78:
of radius 1/2 can accommodate any plane curve of length 1 by placing the midpoint of the curve at the center of the disk. Another possible solution has the shape of a
142:
used a min-max strategy for area of a convex set containing a segment, a triangle and a rectangle to show a lower bound of 0.232239 for a convex cover.
122:
The problem remains open, but over a sequence of papers researchers have tightened the gap between the known lower and upper bounds. In particular,
212:
29:
577:
218:
431:
394:
358:
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306:
Khandhawit, Tirasan; Pagonakis, Dimitrios; Sriswasdi, Sira (2013), "Lower Bound for Convex Hull Area and
Universal Cover Problems",
269:
209:, a set of minimal area that can accommodate every unit-length line segment (with translations allowed, but not rotations)
587:
572:
96:
106:
conjectured that a shape accommodates every unit-length curve if and only if it accommodates every unit-length
203:, the problem of finding a maximum-area shape that can be rotated and translated through an L-shaped corridor
152:
59:
126:
constructed a (nonconvex) universal cover and showed that the minimum shape has area at most 0.260437;
530:
200:
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114:
showed that no finite bound on the number of segments in a polychain would suffice for this test.
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341:
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Gerriets, John; Poole, George (1974), "Convex regions which cover arcs of constant length",
62:
to fit inside the region. In some variations of the problem, the region is restricted to be
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attribute this observation to an unpublished manuscript of
Laidacker and Poole, dated 1986.
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191:. If confirmed, this will reduce the upper bound for the convex cover by about 3%.
356:; Poole, George (2003), "An improved upper bound for Leo Moser's worm problem",
55:
512:
425:
Panraksa, Chatchawan; Wetzel, John E.; Wichiramala, Wacharin (2007), "Covering
547:
445:
372:
329:
215:, find the smallest convex area that can cover any planar set of unit diameter
206:
92:
63:
392:; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser",
47:
26:
What is the minimum area of a shape that can cover every unit-length curve?
221:, find the shortest path to escape from a forest of known size and shape.
102:
It is also not trivial to determine whether a given shape forms a cover.
43:
408:
290:
79:
18:
75:
282:
503:
462:
Wang, Wei (2006), "An improved upper bound for the worm problem",
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308:
International
Journal of Computational Geometry & Applications
58:
of length 1. Here "accommodate" means that the curve may be
51:
183:. Two proofs of the conjecture were independently claimed by
139:
111:
110:
with three segments, a more easily tested condition, but
250:
155:
131:
50:
in 1966. The problem asks for the region of smallest
483:
Panraksa, Chatchawan; Wichiramala, Wacharin (2021),
188:
529:
484:
175:
46:formulated by the Austrian-Canadian mathematician
145:In the 1970s, John Wetzel conjectured that a 30°
531:"Drapeable unit arcs fit in the unit 30° sector"
184:
134:gave weaker upper bounds. In the convex case,
16:Unsolved geometry problem about planar regions
238:
127:
103:
8:
140:Khandhawit, Pagonakis & Sriswasdi (2013)
123:
528:Movshovich, Yevgenya; Wetzel, John (2017),
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444:
407:
371:
319:
159:
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112:Panraksa, Wetzel & Wichiramala (2007)
138:improved an upper bound to 0.270911861.
429:-segment unit arcs is not sufficient",
231:
30:(more unsolved problems in mathematics)
95:cover. Its existence follows from the
251:Norwood, Poole & Laidacker (1992)
213:Lebesgue's universal covering problem
176:{\displaystyle \pi /12\approx 0.2618}
132:Norwood, Poole & Laidacker (1992)
7:
149:of unit radius is a cover with area
135:
432:Discrete and Computational Geometry
395:Discrete and Computational Geometry
359:Discrete and Computational Geometry
486:"Wetzel's sector covers unit arcs"
219:Bellman's lost in a forest problem
14:
270:The American Mathematical Monthly
189:Panraksa & Wichiramala (2021)
491:Periodica Mathematica Hungarica
21:Unsolved problem in mathematics
185:Movshovich & Wetzel (2017)
1:
578:Unsolved problems in geometry
40:mother worm's blanket problem
42:) is an unsolved problem in
239:Gerriets & Poole (1974)
128:Gerriets & Poole (1974)
104:Gerriets & Poole (1974)
54:that can accommodate every
614:
513:10.1007/s10998-020-00354-x
124:Norwood & Poole (2003)
97:Blaschke selection theorem
548:10.1515/advgeom-2017-0011
446:10.1007/s00454-006-1258-7
373:10.1007/s00454-002-0774-3
330:10.1142/S0218195913500076
583:Recreational mathematics
464:Acta Mathematica Sinica
177:
60:rotated and translated
178:
536:Advances in Geometry
153:
36:Moser's worm problem
588:Eponyms in geometry
201:Moving sofa problem
86:Solution properties
409:10.1007/BF02187832
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573:Discrete geometry
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147:circular sector
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108:polygonal chain
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74:For example, a
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38:(also known as
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5:
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542:(4): 497–506,
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497:(2): 213–222,
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470:(4): 835–846,
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439:(2): 297–299,
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402:(2): 153–162,
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366:(3): 409–417,
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314:(3): 197–212,
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76:circular disk
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277:(1): 36–41,
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144:
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118:Known bounds
101:
89:
73:
39:
35:
34:
136:Wang (2006)
56:plane curve
567:Categories
504:1907.07351
261:References
207:Kakeya set
556:125746596
521:225397486
346:207132316
321:1101.5638
168:≈
157:π
48:Leo Moser
195:See also
70:Examples
44:geometry
476:2264090
455:2295060
418:1139077
382:1961007
338:3158583
299:0333991
291:2318909
187:and by
80:rhombus
593:Curves
554:
519:
474:
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380:
344:
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297:
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171:0.2618
93:convex
64:convex
552:S2CID
517:S2CID
499:arXiv
342:S2CID
316:arXiv
287:JSTOR
226:Notes
598:Area
130:and
52:area
544:doi
509:doi
441:doi
404:doi
368:doi
326:doi
279:doi
569::
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540:17
538:,
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495:82
493:,
489:,
472:MR
468:49
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451:MR
449:,
437:37
435:,
414:MR
412:,
398:,
378:MR
376:,
364:29
362:,
340:,
334:MR
332:,
324:,
312:23
310:,
295:MR
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285:,
275:81
273:,
165:12
99:.
66:.
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427:n
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161:/
23::
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