As an extension of the classical John ellipsoid and the \(L_{p}\)-John ellipsoids due to Lutwak–Yang–Zhang, this paper studies (p, q)-John ellipsoids. We consider an optimization problem about the (p, q)-mixed volumes, whose solution is uniquely existed for all \(0<p\le q\). The solution allows us to introduce the concept of (p, q)-John ellipsoids. As applications, we established an analog of the John’s inclusion theorem and Ball’s volume-ratio inequality for (p, q)-John ellipsoids. Moreover, the connection between the isotropy of measures and the characterization of (p, q)-John ellipsoids is demonstrated.

作为对Lutwak-Yang-Zhang的经典John椭球和\（L_ {p} \） - John椭球的扩展，本文研究了（p，  q）-John椭球。我们考虑关于（p，q）混合体积的优化问题， 对于所有\（0 <p \ le q \），其解是唯一存在的。该解决方案使我们能够引入（p，  q）-John椭球的概念。作为应用程序，我们建立了（p，  q）-John椭球的John包含定理和Ball的体积比不等式的类似物。此外，测度的各向同性与（p，  q）-约翰椭球得到了证明。