In 1969, Vic Klee asked whether a convex body is uniquely determined (up to
translation and reflection in the origin) by its inner section function, the
function giving for each direction the maximal area of sections of the body by
hyperplanes orthogonal to that direction. We answer this question in the
negative by constructing two infinitely smooth convex bodies of revolution
about the $x_n$-axis in $\R^n$, $n\ge 3$, one origin symmetric and the other
not centrally symmetric, with the same inner section function. Moreover, the
pair of bodies can be arbitrarily close to the unit ball.