关于复(p, q)-Petty投影体的仿射等周不等式

Affine Isoperimetric Inequality for the Complex (p, q)-Petty Projection Body

摘要: 本文在复空间${ℂ}^n$ 中建立了一类新的仿射等周不等式。通过引入非对称$L_p$ 带状体生成测度，定义了复$\left(p,q\right)$ -Petty投影体，并利用加权平均、Hölder不等式与Jensen不等式，结合实$L_p$ -Petty投影不等式，证明了相应的仿射等周不等式，分别在${\dim }_{ℝ}C=1$ 与${\dim }_{ℝ}C=2$ 两种情形下刻画了等号条件，得到极值体分别为实椭球与Hermite椭球。该结论包含了${ℂ}^n$ 中的$L_p$ -Petty投影不等式。

Abstract: In this paper, we establish a new class of affine isoperimetric inequalities in the complex space ${ℂ}^n$ . By introducing the generating measure of an asymmetric $L_p$ zonoid, we define the complex $\left(p,q\right)$ -Petty projection body, and by means of weighted averaging, Hölder’s inequality, and Jensen’s inequality, together with the real $L_p$ -Petty projection inequality, we prove the corresponding isoperimetric inequality. Furthermore, we characterize the equality conditions in the two cases ${\dim }_{ℝ}C=1$ and ${\dim }_{ℝ}C=2$ , obtaining extremal bodies that are real ellipsoids and Hermite ellipsoids, respectively. This result contains the $L_p$ -Petty projection inequality in ${ℝ}^n$ as a special case.