A Doubly Exponentially Crumbled Cake

Weconsiderthefollowingcakecuttinggame:AlicechoosesasetPofnpointsin thesquare(cake)[0,1]2,where(0,0)∈P;Bobcutsoutnaxis-parallelrectangles withdisjointinteriors,eachofthemhavingapointofPasthelowerleftcorner; Alicekeepstherest. IthasbeenconjecturedthatBobcanalwayssecureatleasthalfofthecake. Thisremainsunsettled,anditisnotevenknownwhetherBobcangetanypositive fractionindependentofn.WeprovethatifAlicecanforceBob’ssharetotend tozero,thenshemustuseverymanypoints;namely,topreventBobfromgaining morethan1/rofthecake,sheneedsatleast22Ω(r) points.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Tobias Christ, Andrea Francke, Heidi Gebauer, Jirí Matousek, Takeaki Uno
• Keywords: AlicechoosesasetPofnpointsin Thesquare, Cake, CORR 2011, Education