Convex bodies with similar projections in the complex spaces.
by
Vladimir Petrovich Golubyatnikov
Sobolev Instritute of Mathematics
Coauthors: Nina Vladimirovna Baltakhinova

We study here the classical problem: Let the projections of two convex bodies in an Euclidean space onto any k -dimensional plane be affinely equivalent. Is it true, that these bodies are affinely equivalent themselves? Which additional conditions guaratnee such an equivalence? See [1-3].

Let h_j be two functions, j = 1, 2 , defined on the unit vectors x in in the complex space Cⁿ , n > 2 , by

h_j (x) = exp ( (-1)^j · < Ax, x > ) .

Here < , > is the Hermitian scalar product, A is an Hermitian matrix whose eigenvalues c_i , i = 1, ... , n satisfy the condition MAX (c_i) - MIN(c_i) < 1/2 . In this case the functions h_j are convex. Let V_j be convex centrally symmetric bodies in Cⁿ with the support functions h_j . We prove that for any 2-dimensional COMPLEX plane in Cⁿ the projections of the bodies V_j onto this plane after anti-conformal transformation (z₁ , z₂ ) --> ( - y₂ , y₁ ) in any orthonormal basis e₁ , e₂ of this plane become homothetic with the homothety coefficient exp ( < A e₁, e₁ > + < A e₂, e₂ > ) , here y_j is the complex conjugate to z_j . For the real case, this construction was condsidered earlier in [2, 3].

It follows from the results of [2] that the bodies V_j are affinely equivalent in Cⁿ if and only if c_i + c_n+1-i = a for some constant a , i = 1, ... , n .

REFERENCES

1. Golubyatnikov, V.P. On unique recoverability of convex and visible bodies from their projections I. Mathem. USSR Sbornik, 1991, v. 73, 1 - 10.

2. Gardner, R.J., Vol^ci^c, A. Convex bodies with similar projections. Proc. AMS. 1994, v.121, 563 - 568.

3. Petty, C.M., Mc-Kinney, J.R. Convex bodies with circumscribing boxes of constant volume. Portug. Math., 1987, v. 44, 447 - 455.

Date received: January 14, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadw-02.