Multiple integration of UnitStep fails

*To*: mathgroup at smc.vnet.net*Subject*: [mg63194] Multiple integration of UnitStep fails*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>*Date*: Sat, 17 Dec 2005 03:46:34 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Hello group, trying to solve the nice problem of determining the probability pn that a polygon formed by n (>=4) random points on the unit circle is void of an acute angle I came up with the following multiple integral (written down here for n=5) In[15]:= p5 = (4!*Integrate[Integrate[Integrate[Integrate[UnitStep[\[Phi]4 - Pi, \[Phi]5 - \[Phi]2 - Pi, Pi - \[Phi]3, Pi - \[Phi]4 + \[Phi]2, Pi - \[Phi]5 + \[Phi]3], {\[Phi]5, \[Phi]4, 2*Pi}], {\[Phi]4, \[Phi]3, 2*Pi}], {\[Phi]3, \[Phi]2, 2*Pi}], {\[Phi]2, 0, 2*Pi}])/(2*Pi)^4 Mathematica version 4 was not able to solve this but returned it unevaluated after some minutes; version 5 complained several things like: argument is not a power series, unable to check convergence, but didn't come up with any result in ten minutes (I wouldn't wait longer). I could successfully check the normalization at least: In[14]:= p5 = (4!*Integrate[Integrate[Integrate[Integrate[UnitStep[1], {\[Phi]5, \[Phi]4, 2*Pi}], {\[Phi]4, \[Phi]3, 2*Pi}], {\[Phi]3, \[Phi]2, 2*Pi}], {\[Phi]2, 0, 2*Pi}])/(2*Pi)^4 Out[14]= 1 How would you proceed to solve In[15]? What about the general case (n=6 see below)? Any hints are greatly appreciated. Regards, Wolfgang PS: Here's the probability for the case n=6 p6 = (5!*Integrate[Integrate[Integrate[Integrate[Integrate[UnitStep[\[Phi]5 - Pi, \[Phi]6 - \[Phi]2 - Pi, Pi - \[Phi]3, Pi - \[Phi]4 + \[Phi]2, Pi - \[Phi]5 + \[Phi]3, Pi - \[Phi]6 + \[Phi]4], {\[Phi]6, \[Phi]5, 2*Pi}], {\[Phi]5, \[Phi]4, 2*Pi}], {\[Phi]4, \[Phi]3, 2*Pi}], {\[Phi]3, \[Phi]2, 2*Pi}], {\[Phi]2, 0, 2*Pi}])/(2*Pi)^5 $Aborted