An upper bound for permanents of nonnegative matrices

A recent conjecture of Caputo, Carlen, Lieb, and Loss, and, independently, of the author, states that the maximum of the permanent of a matrix whose rows are unit vectors in l p is attained either for the identity matrix I or for a constant multiple of the all-1 matrix J. njecture is known to be true for p = 1 (I) and for p ⩾ 2 (J). ve the conjecture for a subinterval of ( 1 , 2 ) , and show the conjectured upper bound to be true within a subexponential factor (in the dimension) for all 1 < p < 2 . In fact, for p bounded away from 1, the conjectured upper bound is true within a constant factor.