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We give the sharp lower bound of the volume product of |$n$| -dimensional convex bodies that are invariant under the discrete subgroups |$SO(K)=\{ g \in SO(n); g(K)=K \}$|⁠ , where |$K$| is an |$n$| -cube or |$n$| -simplex. This provides new partial results of Mahler's conjecture and its nonsymmetric version. In addition, we give partial answers for Viterbo's isoperimetric-type conjecture in symplectic geometry from the view point of Mahler's conjecture. [ABSTRACT FROM AUTHOR]

We are interested in the asymptotic behavior of ground states for a class of quasilinear elliptic equations in R 3 when the nonlinear term has H 1 -critical growth. In the previous result [Adachi et al. Asymptotic property of ground states for a class of quasilinear Schrödinger equation with H 1 -critical growth. Calc Var Partial Differential Equations. 2019;58(3). Art. 88, 29 pp.], it was shown that, after a suitable scaling, the ground state converges to the Talenti function. However, the uniqueness of the limit of the full sequence was not obtained, which was essentially owning to the fact that the Talenti function does not belong to L 2 (R 3). In this paper, by constructing a refined test function and performing a detailed asymptotic analysis, we are able to obtain the uniqueness of asymptotic limit of ground states. [ABSTRACT FROM AUTHOR]

We are interested in the asymptotic behavior of ground states for a class of quasilinear elliptic equations in R 3 when the nonlinear term has H 1 -critical growth. In the previous result [Adachi et al. Asymptotic property of ground states for a class of quasilinear Schrödinger equation with H 1 -critical growth. Calc Var Partial Differential Equations. 2019;58(3). Art. 88, 29 pp.], it was shown that, after a suitable scaling, the ground state converges to the Talenti function. However, the uniqueness of the limit of the full sequence was not obtained, which was essentially owning to the fact that the Talenti function does not belong to L 2 (R 3). In this paper, by constructing a refined test function and performing a detailed asymptotic analysis, we are able to obtain the uniqueness of asymptotic limit of ground states. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type rearrangement inequality based on the Steiner rearrangement. To show orbital stability of standing wave solutions with respect to a nonlinear Schrödinger equation, $$H^1$$ -precompactness of minimizing sequence of $$L^2$$ -constraint minimizing problem is very important. Usually, by using the concentration compactness principle and some scaling argument, $$H^1$$ -precompactness is obtained. To prove $$H^1$$ -precompactness without scaling arguments, we introduce the rearrangement. The Steiner rearrangement is defined as a map from $$H^1$$ to $$H^1$$ , whereas our rearrangement $$(\cdot \star \cdot )$$ is defined as a map from $$H^1 \times H^1$$ to $$H^1$$ . By using the rearrangement, we show a strict inequality $$\Vert \nabla (u \star v) \Vert _{L^2}^2 < \Vert \nabla u\Vert _{L^2}^2 + \Vert \nabla v\Vert _{L^2}^2$$ under simple assumptions. [ABSTRACT FROM AUTHOR]