Subject: Question to topology list From: Thomas Schick Date: Thu, 27 Oct 2005 13:52:24 +0200 (CEST) Dear Amit, it seems to me that the proposition you hope for isn't true in general (not even in dimension 2). Here is my sketch for a counterexample: take the square [0,10]\times [0,10] in R2. We need two functions f,g which are strictly quasiconcave for the first and the second coordinate direction according to your definition. write f(x,y)=f_y(x) as a family of functions on [0,10]; each of them has to be quasiconcave. Define the f_y to be piecewise linear, each of them being equal to f_y(x)=x on [0,1], and then extend continuously by a linear function of slope -1/9- 1000*y*(1-y). In any case, for y=0 and y=1, the function goes just goes down to 0 on [0,10], whereas in the middle it is monotonously decreasing to something much smaller. Certainly this is strictly quasiconcave. Given any subset X of the square which contains 0\times [0,10], B_1 will produce the union of X and B_1(0\times [0,10])=:A_1, the latter one containing the three sides 0\times [0,10], [0,10\times 0 and [0,10]\times 1, but not the point (5,5) neither a neighborhood of (5,5) (this follows because the functions f_y(x) are monoton decreasing for x>1, and more precisely A_1 is a halfmoonshaped thickening of these three sides). Define g(x,y)=f(y,x). Everything said about f applies to g, if we interchange the roles of x and y. In particular, B_2([0,10]\times 0)=:A_2 contains [0,10]\times 0, 0\times [0,10], 1\times [0,10] but not an neighborhood of (5,5) Now start with a=(0,0) and form S^m. S2 = [0,10]\times 0, then S3=A_2, S4 is the union of A_2 and A_1, the same for S5,... S_bar seems to be the closure of A_1 union A_2, which radially retracts onto the boundary of the square. A composition of this retract with a rotation of the boundary is then a self map without fixed points. I think the retraction is actually a deformation retraction, so the map is homotopic to the identity. Is there a mistake I'm making, or do you have additional conditions in your 2-dimensional theorem? Best regards, Thomas Schick ----------------------------------------------------------- Thomas Schick | email: schick@uni-math.gwdg.de Bunsenstr. 3 | phone: ++49 551 397766 37073 Goettingen | fax: ++49 551 392985 Germany | http://www.uni-math.gwdg.de/schick -----------------------------------------------------------