In this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke– Santal´o type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.

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Given a closed exact Lagrangian in the cotangent bundle of a closed smooth manifold, we prove that the projection to the base is a simple homotopy equivalence.

Denote by Mnthe set of n ×ncomplex matrices. Let f:Mn→[0, ∞)be a continuous map such that f(μUAU∗) =f(A)for any complex unit μ, A ∈Mnand unitary U∈Mn, f(X) =0if and only if X=0and the induced map t →f(tX)is monotonically increasing on [0, ∞)for any rank onenilpotent X∈Mn. Characterization is given for surjective maps φon Mnsatisfying f(AB−BA) =f(φ(A)φ(B) −φ(B)φ(A)). The general theorem isthen used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius.

We study how conserved quantities such as angular momentum and center of mass evolve with respect to the retarded time at null infinity, which is described in terms of a Bondi-Sachs coordinate system. These evolution formulae complement the classical Bondi mass loss formula for gravitational radiation. They are further expressed in terms of the potentials of the shear and news tensors. The consequences that follow from these formulae are (1) Supertranslation invariance of the fluxes of the CWY conserved quantities. (2) A conservation law of angular momentum \`a la Christodoulou. (3) A duality paradigm for null infinity. In particular, the supertranslation invariance distinguishes the CWY angular momentum and center of mass from the classical definitions.