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M ahler's theory of irreducible star bodies is redeveloped and extended in a modified form. It is shown that any closed bounded star set S contains a closed irreducible star set T having the same critical determinant. Further, it is shown that, if the first set S is bounded by a finite number of algebraic surfaces, then there will be an irreducible set T which is also bounded by a finite number of algebraic surfaces.doi:10.1098/rsta.1952.0017 fatcat:pyyxendxfzb6jedhnkfsy4jlny