Böröczky Károly:
A "kup-terfogat" mertekrol
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One of the cornerstones of the Brunn-Minkowski theory of convex bodies is
the Minkowski problem about the existence of a convex body with prescribed
surface area measure. Here we are concerned with the Minkowski problem for
the cone volume measure, whose study was iniated by Gromov and Milman. For
an origin symmetric convex body in $R^d$ of volume one, its cone volume
measure is the probability measure on the sphere $S^{d-1}$, which is the
integral of the normalized support function with respect to the surface 
area
measure. For an origin symmetric convex polytope, it is the discrete
probability measure where the measure of an exterior unit normal is the
volume of the cone over the corresponding facet.
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In this talk, the characterization of the cone volume measures
among even probability measures on $S^{d-1}$ is presented.
In addition, the uniqueness problem is discussed in the plane and
for unconditional convex bodies. 
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This is joint work with E. Lutwak,
D. Yang and G. Zhang.