The Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

Let Mt be an isoparametric foliation on the unit sphere (Sn−1(1), gst) with d principal curvatures. Using spherical coordinates induced by Mt, we construct a Minkowski norm representation $$F = r\sqrt {2f(t)} $$ , which generalizes notions of (α, β)-norm and (α1, α2)-norm. technique local frame, give exact explicit answer to question when really defines norm. similar technique, study Hessian isometry Φ between two norms preserves orientation fixes ξ-coordinates. There are ways describe this Φ, either system ODEs, or its restriction any normal plane for is then reduced ℝ2 satisfying certain symmetry (d)-properties. When > 2, prove that can obtained gluing positive scalar multiplications compositions Legendre transformation multiplications, so it must satisfy (d)-property orthogonal decomposition ℝn V′ + V″, i.e., nonzero x x′ x″ $$\Phi (x) \bar x\prime \prime $$x\prime ,\bar \in {\bf{V}}\prime x″, have $$g_x^{{F_1}}(x\prime ,x) g_{\bar x}^{{F_2}}(\bar x)$$ . As byproducts, following results. On indicatrix (SF, g), where F g metric, Nt SF ∩ ℝ>0M0 isoparametric. Laugwitz Conjecture valid if metric flat ℝn{0} n Euclidean.