Volume of points satsfying some inequality/equality constraints
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From: Golabi Doon on 24 Dec 2009 06:47 Hello folks, Let's denote a point in R^n by (x_1,...x_n). I have to compute the volume of points in R^n which satsify the following constraints: x_i >= 0 sum_{i=1}^n x_i = 1 sum_{i=1}^n x_i*log(x_i) >= -a where a is a given constant such that 0<=a<=log(n), and x*log(x) at x=0 is defined to be 0. Do you think if there is a closed form solution for this volume in terms of "n" and "a"? If not, even an upper bound on this volume would be of great help. Your comments would be highly appreciated. Golabi
From: Mike Terry on 24 Dec 2009 09:41 "Golabi Doon" <golabidoon(a)gmail.com> wrote in message news:38ea960f-501b-4907-8aa4-5771a07fb799(a)d10g2000yqh.googlegroups.com... > Hello folks, > > Let's denote a point in R^n by (x_1,...x_n). I have to compute the > volume of points in R^n which satsify the following constraints: > x_i >= 0 > sum_{i=1}^n x_i = 1 Straight away, it's clear the volume will be zero, as the last constraint restricts the points to lying on some n-1 dimensional hyper-plane within the n dimensional space. > sum_{i=1}^n x_i*log(x_i) >= -a > > where a is a given constant such that 0<=a<=log(n), and x*log(x) at > x=0 is defined to be 0. > > Do you think if there is a closed form solution for this volume in > terms of "n" and "a"? > > If not, even an upper bound on this volume would be of great help. > > Your comments would be highly appreciated. > > Golabi
From: Golabi Doon on 24 Dec 2009 23:58
On Dec 24, 8:41 am, "Mike Terry" > Straight away, it's clear the volume will be zero, as the last constraint > restricts the points to lying on some n-1 dimensional hyper-plane within the > n dimensional space. I meant the n-1 dimensional volume, such as the length of a line segment in plane or a bounded cut of a plane in 3d space. Sorry about the confusion. Golabi |