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In 1969, Vic Klee asked whether a convex body is uniquely determined (up totranslation and reflection in the origin) by its inner section function, thefunction giving for each direction the maximal area of sections of the body byhyperplanes orthogonal to that direction. We answer this question in thenegative by constructing two infinitely smooth convex bodies of revolutionabout the $x_n$-axis in $\R^n$, $n\ge 3$, one origin symmetric and the othernot centrally symmetric, with the same inner section function. Moreover, thepair of bodies can be arbitrarily close to the unit ball.

A problem of Klee on inner section functions of convex bodies