ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 20, 43, 52 Iwasaki, Katsunori, Kyushu University, Japan Polyhedral harmonic functions and invariant theory The harmonic functions are precisely those continuous functions which satisfy the mean value property with respect to a ball. What happens if the ball is replaced by a polytope? Our main theorem states that for any ndimensional polytope P, the set H_{P} of all continuous functions satisfying the mean value property with respect to P forms a finite-dimensional linear space of polynomials. The problem of characterizing the function space H_{P} is of particular interest when P admits ample symmetry. Then the invariant theory for the symmetry group of P serves as a powerful tool to determine H_{P} explicitly. Several examples are also given. Reference: K. Iwasaki, Polytopes and the mean value property, Discrete & Comput. Geometry 17 (1997), 163–189.