In this paper we study the L p-Minkowski problem for p=鈭n鈭1, which corresponds to the critical exponent in the Blaschke鈥揝antalo inequality. We first obtain volume estimates for general solutions, then establish a priori estimates for rotationally symmetric solutions by using a Kazdan鈥揥arner type obstruction. Finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow-up analysis. We also include an existence result for the L p-Minkowski problem which corresponds to the super-critical case of the Blaschke鈥揝antalo inequality.

abstract

We construct a special principal series representation for the modular double Uqq∼(gR) of type Ar representing the generators by positive essentially self-adjoint operators satisfying the transcendental relations that also relate q and. We use the cluster variables parameterization of the positive unipotent matrices to derive the formulas in the classical case. Then we quantize them after applying the Mellin transform. Our construction is inspired by the previous results for gR =sl(2,R) and can be generalized to all other types of simple split real Lie algebra. We conjecture that our positive representations are closed under the tensor product and we discuss the future perspectives of the new representation theory following the parallel with the established developments of the finite-dimensional representation theory of quantum groups.

Using the complex coloring method, we present the graphs of the quantum dilogarithm function Gb(z) and visualize its analytic and asymptotic behaviours. In particular we demonstrate the limiting process when the modified Gb(z)→Γ(z) as b→0. We also survey the relations of Gb(z) with different variants of the quantum dilogarithm function.

We construct the positive principal series representations for Uq(gR) where g is of simply-laced type, parametrized by Rr where r is the rank of g. In particular, the positivity of the operators and the transcendental relations between the generators of the modular double are shown. We define the modified quantum group $\mathbf{U}_{q\tilde{q}(g_R)$ of the modular double and show that the representation of both parts of the modular double commute with each other, there is an embedding into the q-tori polynomials, and the commutant is the Langlands dual. We write down explicitly the action for type An,Dn and give the details of calculations for type E6,E7 and E8.