ABSTRACT
Let H {u) be a positively homogeneous function of degree one in vectors u in 3-dimensional space. Let H be the envelope of the family of planes ux = H(u). Its parametric equation is x\ = x2 — X3 = The eigenvalues Ri and R 2 of the second differential d2 H(u) for unit vectors ii are the principal radii of curvature of the surface H . [/£] and R 2 may be zero. We disregard trivial zero eigenvalues of d?H(u) [1].]
T h e o r e m 1. Let f ( R \ , R i, n), a piecewise analytic function o f R \ and R i, the points n being on the unit sphere, be defined in a domain D such that R i ^ R i. Let and have like signs everywhere. I f there exists a piecewise analytic function II(it) with eigenvalues R j and R 2 for its second differential belonging to the domain D such that
f ( R i ,R 2 ,n ) = g(h),
where g(n) is a given function o f n, then H ( tt) is unique up to a linear term au, i.e., the corresponding surface H is unique up to a translation.
