The equilibrium shape of a crystal is the celebrated Wulff shape W, which can be obtained via the Wulff flow. When the initial crystal K is convex, the image under the Wulff flow is K + tW after time t. The n + 1-dimensional volume V_n+1(K + tB) is the well known Steiner polynomial , where B is the closed unit ball....[ Read more ]

The equilibrium shape of a crystal is the celebrated Wulff shape W, which can be obtained via the Wulff flow. When the initial crystal K is convex, the image under the Wulff flow is K + tW after time t. The n + 1-dimensional volume V_n+1(K + tB) is the well known Steiner polynomial , where B is the closed unit ball.

Wulff flow and Wulff shape has been highly studied in crystal growth. Ge-ometric information encoded in the the associated Wulff-Steiner polynomial was far from being thoroughly understood.

In this thesis, Wulff-Steiner polynomial of degree 2 was surveyed. We ob-tained a characterization of them. The relative positions among the roots , the W-inradius, W-outradius and other geometric quantities are studied. Wulff-Steiner polynomial have no positive real root by its definition. For degree 2, the fact that it must have negative real roots is equivalent to the classi-cal isoperimetric inequality. For degree 3, we showed that complex roots can occur and that the real parts of any complex root must be negative. As an application of the Wulff-Steiner polynomial, we found an invariant under the Wulff flow in any dimension n ≥ 2.