Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we prove a local trace formula for the Ginzburg-Rallis model. By applying this trace formula, we prove the multiplicity one theorem for the Ginzburg-Rallis model over the tempered Vogan L-packets. In some cases, we also prove the epsilon dichotomy conjecture which gives a relation between the multiplicity and the exterior cube epsilon factor. This is a sequel work of [Wan15] in which we proved the geometric side of the trace formula.

We study a local multiplicity problem related to so-called generalized Shalika models. By establishing a local trace formula for these kind of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are constant over every discrete Vogan L-packet and are related to local exterior square L-functions.

TheLpAleksandrovintegralcurvatureanditscorrespondingchar- acterization problem—the Lp Aleksandrov problem—were recently introduced by Huang, Lutwak, Yang, and Zhang. The current work presents a solution to the Lp Aleksandrov problem for origin-symmetric polytopes when −1 < p < 0.

Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer’s curvature measures within the dual Brunn-Minkowski theory and stated the “Minkowski problem” associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is 0) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space)—two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by B ̈or ̈oczky, Henk & Pollehn that these new sufficiency conditions are also necessary.

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