Abstract: This paper defines a new Sylvester type functional A(K). Using the method of shadow systems, the extremal problems of A(K) are studied. When K is a ellipsoid, A(K) attains its minimum value. In a plane, A(K) attains its maximum value when Kis a triangle. When K is symmetric, the corresponding example is parallelogram. 

Key words: Orlicz centroid body, shadow system, Sylvester type functional