The Heilbronn Problem for Convex Regions

In this variant, the enclosing container is itself optimized: the points may lie in any convex region of unit area. Because any area outside the points' convex hull only decreases the normalized ratio, the optimal container is always the convex hull of the point set itself. The objective A is thus the minimal triangle area divided by the convex hull area. Optimality has been established for n ≤ 8.

A = $1$ = 1

Trivial.

3-fold dihedral symmetry · 1 minimal triangle.

details & viewer · coordinates proven

A = $\frac{1}{2}$ = 0.5

Trivial.

4-fold dihedral symmetry · 4 minimal triangles.

details & viewer · coordinates proven

A = $\frac{5 - \sqrt{5}}{10}$ = 0.2763932022500210…

Found by David Cantrell, June 2007.

Proved optimal by Tej Stead, August 2026.

5-fold dihedral symmetry · 5 minimal triangles.

details & viewer · coordinates proven

A = $\frac{1}{6}$ = 0.1666666666666666…

Found by David Cantrell, June 2007.

Proved optimal by Andreas Dress, Lu Yang, and Zhenbing Zeng, 1995.

6-fold dihedral symmetry · 6 minimal triangles.

details & viewer · coordinates proven

A = $\frac{1}{9}$ = 0.1111111111111111…

Proved optimal by Zhenbing Zeng and Lu Yang, 1995.

120° rotationally symmetric · 9 minimal triangles.

details & viewer · coordinates proven

A = 0.0800001393294664…

Root of $2060 x^{5} - 2332 x^{4} + 1064 x^{3} - 240 x^{2} + 26 x - 1$ .

Found by David Cantrell, June 2007.

Proved optimal by Tej Stead, September 2026.

Mirror symmetric · 10 minimal triangles.

details & viewer · coordinates proven

A = 0.0640647523079416…

Root of $4636072741330 x^{14} + 1774549666005 x^{13} - 1233481443979 x^{12} - 1367692325626 x^{11} - 694825007492 x^{10} + 38268929375 x^{9} + 102061390459 x^{8} + 123696372 x^{7} - 5255868622 x^{6} + 66844595 x^{5} + 131075035 x^{4} - 8755802 x^{3} - 171936 x^{2} - 24711 x + 2645$ .

Found by David Cantrell, June 2007.

Mirror symmetric · 11 minimal triangles.

details & viewer · coordinates record

A = 0.0519930715229432…

Root of $9088 x^{4} - 4928 x^{3} + 708 x^{2} - 44 x + 1$ .

Found by David Cantrell, June 2007.

2-fold dihedral symmetry · 16 minimal triangles.

details & viewer · coordinates record

A = $\frac{2}{47}$ = 0.0425531914893617…

Found by David Cantrell, June 2007.

Mirror symmetric · 28 minimal triangles.

details & viewer · coordinates record

A = $\frac{2}{51}$ = 0.0392156862745098…

Found by David Cantrell, June 2007.

3-fold dihedral symmetry · 36 minimal triangles.

details & viewer · coordinates record

A = 0.0309372079216206…

Found by David Cantrell, June 2007.

Not symmetric · 19 minimal triangles.

details & viewer · coordinates record

A = 0.0278355805175536…

Found by AlphaEvolve, June 2025.

Not symmetric · 21 minimal triangles.

details & viewer · coordinates record

A = 0.0245640556132301…

Found by Tej Stead, July 2026.

Not symmetric · 25 minimal triangles.

details & viewer · coordinates record

A = 0.0222728771289812…

Root of $4608 x^{5} - 10752 x^{4} + 9824 x^{3} - 3712 x^{2} + 482 x - 9$ .

Found by David Cantrell, June 2007.

4-fold dihedral symmetry · 28 minimal triangles.

details & viewer · coordinates record

A = 0.0190573327521721…

Found by Rhys Chappell, September 2026.

Not symmetric · 28 minimal triangles.

details & viewer · coordinates record

A = $\frac{2 - \sqrt[3]{\frac{3}{2} - \frac{\sqrt{\frac{239}{3}}}{6}} - \sqrt[3]{\frac{3}{2} + \frac{\sqrt{\frac{239}{3}}}{6}}}{18}$ = 0.0182388954634910…

Found by Tej Stead, July 2026.

6-fold dihedral symmetry · 36 minimal triangles.

details & viewer · coordinates record

A = 0.0159310923501429…

Root of $16654996327987663164080307837682632049874839205015076556456543273852809727444121317152538714517183919 x^{51} - 50741002518352730600264528339104376773434838790202632540805147086034247252763865792753920464267635573 x^{50} + 59523505893012158559437235867853244947179692023685017476500188651227296076109329215656086064011267489 x^{49} - 36987041733778503631684570054578915904064764481393835129403048466929628726892550253194595633252726850 x^{48} + 13802020942359781359532835670331400292457190606793717904934340951894516144229404778573568979157901556 x^{47} - 3296658808903291667381651953620346122872866425015747469513998802768001079781870454051426962094273277 x^{46} + 522699078360979182096354950715984808762864042609018058610467299133481973843165505430892133091818631 x^{45} - 56082102368193268784691303908377332071558230085164345974612381847171956162447511292919077809772193 x^{44} + 3779588158414745759040253926156411201169394223850594499546797341163372099615339644694043803476810 x^{43} + 38804158227810064872881847143965708389640299045318202795044060326916513806603210230514813480670 x^{42} - 76245308599054938450279905392109547756658057869894850434217737857123867652222317431472472454500 x^{41} + 17213914510096799610910794433407976213793308524909473858407512374332360333733441195768093958666 x^{40} - 2581698546922737206754552938808007819834442770590777748460326465749984379596151951704067940505 x^{39} + 312072360548528036615085784780657382510240660922647503094597198643874680225097890033187808289 x^{38} - 34094820277978157885246445053088308041049796551224894512623542089173869593754653498973936641 x^{37} + 3410402811145246131813770717309440263360780712183061972564114609654437120334322647184416544 x^{36} - 271282760586160591069969069140476945510429502380734626569881260277916827821253614589450353 x^{35} + 8964213447929428305402665467314421858263171553663157732197428154274832533890558641276837 x^{34} + 1923203007732281067391499795416781297576812695277433611936479858013445273721466583889417 x^{33} - 501518281647971586746624890890317734768769703924228123088295643341747807703660169964670 x^{32} + 74901741414760772803431635718001969326647968436687168820562144741951877663763333348762 x^{31} - 8793339854703183362216942014089270298583083848430875674639726423156497665143583856599 x^{30} + 884440462041187107283475346039472588977291523754286257033259570465574394383577964060 x^{29} - 78974785991390863296101086854453347114231599106556602496429768566048683284604195488 x^{28} + 6351440938225235091307229476747584664822770409807711680636450788763662338660730736 x^{27} - 461577386020991827608535129901123426274312148150149667208924550289794040921123158 x^{26} + 30241062227720993053683464944230198235675014316896137072924910087824613616621533 x^{25} - 1777889704731817957933477778233549802357792023928514745543045283724426249599528 x^{24} + 93267061234360602967296674081132845199112348187406824265727268012016566200583 x^{23} - 4338371522575923233034210097677980674235870009879661484100060767182893535897 x^{22} + 177640264174182468779901368262860951792277130596752158820640835566528124760 x^{21} - 6346615055201503020261343318716637981874577322541383923515904824683154528 x^{20} + 195719278657162866559477668957744832524686044349121546932247027308617521 x^{19} - 5146798111408492011733625064073866364135976197359254613756039953565431 x^{18} + 114342597713325086360466620824031491178078220012926733476841878296701 x^{17} - 2166532041106191143395488189771983665844918905009609494849319862555 x^{16} + 37534499535017718712938026546492630945885931493825986952019257529 x^{15} - 686311606973090058567849302474565048555704337440088851152041666 x^{14} + 13463640572723940217486049731712260212765015277098533721414827 x^{13} - 208766904360611863076421478201045069313767677355624072803244 x^{12} + 523821011921006477340257754559207277970409454884057162450 x^{11} + 85093250513592250373365314115539360225677251028427399015 x^{10} - 2714743802786573644617033845589881585012310918640093409 x^{9} + 45990737626797492919030822727366787337965820963078068 x^{8} - 487271946585283985377454739508037532093635633664116 x^{7} + 2432093267720660392912770667937619993847163656888 x^{6} + 5979744051687564902194282372076326165161580304 x^{5} + 4273919079255637923529042315667407153215968 x^{4} - 892839406092143487318163123564942857358272 x^{3} - 5635659728305912723837243683640292006016 x^{2} - 1029008599106478248990748958653643776 x + 87191354287695850324598344835925504$ .

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 33 minimal triangles.

details & viewer · coordinates record

A = 0.0144642050306691…

Root of $38125 x^{6} + 304000 x^{5} + 238650 x^{4} - 101700 x^{3} - 31260 x^{2} + 4690 x - 61$ .

Found by Nathan Sudermann-Merx, August 2026.

5-fold dihedral symmetry · 40 minimal triangles.

details & viewer · coordinates record

A = 0.0131910874541530…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

120° rotationally symmetric · 39 minimal triangles.

details & viewer · coordinates record

A = 0.0122549627897763…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 38 minimal triangles.

details & viewer · coordinates record

A = 0.0110824852805632…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 38 minimal triangles.

details & viewer · coordinates record

A = 0.0107126406526939…

Root of $397777245318657641320538922492702173952372763389389043694980360764148997791 x^{72} - 2716620312646442145010329127161073018234118866271499021149062332236475422355 x^{71} + 9419238705687373592093473464202689405045883184265793655568319891909814648529 x^{70} - 21363559330289112405526772870010341073826778349323518609930264029437609525985 x^{69} + 34922553732538168330320775202965368380111520750791423287634157820473364142040 x^{68} - 43463500814547433326200680827812277790675600149828727395331496847005029976757 x^{67} + 42762092455077781469974130900789698235184710865471422742196060252943413940169 x^{66} - 34207814063061502743709291775216222413487113175613793080911469655305001203810 x^{65} + 22743450828039780442490178278158233769963102225819546113850930779893488303890 x^{64} - 12790556131402495427934990318865807806703013256359443957966601572892199273318 x^{63} + 6172973744289483501261716515449892230471495200712868270308211486663117920378 x^{62} - 2588159085704047919861241280848496560196123937655520803064849798046593516953 x^{61} + 953122342638692898472219741681118482514637334387072445062321604437686997437 x^{60} - 311640843008317992853866796378943034535416516567188374933402930213847826426 x^{59} + 91563675508150054810109770002265510679615570148831251773152499418751024579 x^{58} - 24539623530518244827109247789030550326840819114793650796911975034041300553 x^{57} + 6117978518660378680353492588801081426562890503925033093450081125711626416 x^{56} - 1453411092461016726785484718781861670725755990616793729372143282954833092 x^{55} + 336954068454701221549376197824873369950100800810524811451891546660174900 x^{54} - 77347285076717518615452896134159764589941968436555078692531420262615623 x^{53} + 17542746271945369131999007436694149164156578142755584425773009143156560 x^{52} - 3867043687263816233277545255489000307628677245183327326380483776948859 x^{51} + 810725491303092456773931368104030179634457538756204507999184328417302 x^{50} - 158534713109751283348588878443963141268135631299282603395101145566665 x^{49} + 28480935863285876677160383060670035664932176478938473084928768136777 x^{48} - 4643389698098243908306276087735052373330246326802532587487487337615 x^{47} + 678578602876838205416281380674070976528808855551598058880303290202 x^{46} - 87421520179259343760603480686435194198480366042730984338377618707 x^{45} + 9646738564008262320947758588019710572132701065455957839577488209 x^{44} - 855769033429430904297725287948530983410948666509364557667600152 x^{43} + 49518462643018888988448982529079717435903138996755573454076447 x^{42} + 748451296285745102069006654462173402536295838731052706137118 x^{41} - 699057640410458581591382207070495473376941247673729336498724 x^{40} + 115610992047326666723331646523292772859499526824076804113993 x^{39} - 12638733371508100778612358130644295756168175094406087297932 x^{38} + 985220526536407929967062475625769458393437052077012873604 x^{37} - 45773343410408087064826165186217409661264701566385736361 x^{36} - 846167375175331975510253352138336194754240809216816562 x^{35} + 446718775658654635474382890518409228731059955894493178 x^{34} - 56556078128055014118455338011176259201411800931595426 x^{33} + 5012860743832344043730931429909056193476716505066893 x^{32} - 356623280635903031119857155364721862006630424747125 x^{31} + 20629598232991049235282616056194233821785787435973 x^{30} - 832968883643296327906673132401712278727829534470 x^{29} + 222552108952131024700483377620778479285074025 x^{28} + 3792287476409091730072438757309046957135741603 x^{27} - 384471930777311650091077793274173906797649010 x^{26} + 21574140985758085719380148285322981089512634 x^{25} - 700166703842012814142173085063281742703552 x^{24} + 15108554647318290324980212311406663279723 x^{23} - 1092052821316786026782625249313066823786 x^{22} + 92053083060887357716436740707541412945 x^{21} - 1279257840451294718129076541434977205 x^{20} - 333127863609644227505836414831572123 x^{19} + 25101609723681746182669234366017910 x^{18} - 684261038229017093272789787365479 x^{17} + 12973536723168598964128641312171 x^{16} - 1803428991463021218267596633604 x^{15} + 138407362454210616137358761696 x^{14} - 3510237864164149092691696852 x^{13} - 69266602678385283568630063 x^{12} + 6035926628328881219403562 x^{11} - 106473004460484424702413 x^{10} + 953350610725696299608 x^{9} - 263807465683231568900 x^{8} + 23980367108587283440 x^{7} - 1107535648797423520 x^{6} + 32958528333527712 x^{5} - 672227047557184 x^{4} + 9569009605632 x^{3} - 97173328640 x^{2} + 668183040 x - 2236416$ .

Found by Nathan Sudermann-Merx, August 2026.

3-fold dihedral symmetry · 48 minimal triangles.

details & viewer · coordinates record

A = 0.0093161059776902…

Found by Rhys Chappell, August 2026.

Not symmetric · 45 minimal triangles.

details & viewer · coordinates record

A = 0.0087147876362939…

Found by Rhys Chappell, August 2026.

Not symmetric · 46 minimal triangles.

details & viewer · coordinates record

A = 0.0080291217784917…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Not symmetric · 48 minimal triangles.

details & viewer · coordinates record

A = 0.0077188404339415…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

2-fold dihedral symmetry · 48 minimal triangles.

details & viewer · coordinates record

A = 0.0072891153274598…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Not symmetric · 52 minimal triangles.

details & viewer · coordinates record

A = 0.0071601845302816…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

5-fold dihedral symmetry · 55 minimal triangles.

details & viewer · coordinates record

A = 0.0063096707080895…

Found by Alexandar Lackovic with help of Opus 5.5, September 2026.

Not symmetric · 55 minimal triangles.

details & viewer · coordinates record

A = 0.0058507143333531…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

2-fold dihedral symmetry · 56 minimal triangles.

details & viewer · coordinates record

A = 0.0054701360672898…

Found by Alexandar Lackovic with help of Opus 5.5, September 2026.

Not symmetric · 61 minimal triangles.

details & viewer · coordinates record

A = 0.0052062169865145…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

2-fold dihedral symmetry · 62 minimal triangles.

details & viewer · coordinates record

A = 0.0047738053311543…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

5-fold dihedral symmetry · 70 minimal triangles.

details & viewer · coordinates record