Related papers: Valuations and Surface Area Measures

We define a GL-variety to be a (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used…

In this paper, we investigate an $L_{p}$ Christoffel-Minkowski-type problem that prescribes a class of $L_p$ geometric measures, which are mixtures of the $k$-th area measure and the $q$-th dual curvature measure. By establishing a gradient…

This paper is devoted to exploring the relationship between the $[1,n)\ni p$-capacity and the surface-area in $\mathbb R^{n\ge 2}$ which especially shows: if $\Omega\subset\mathbb R^n$ is a convex, compact, smooth set with its interior…

We exhibit some nontrivial evaluations of the areal Mahler measure of multivariable polynomials, defined by Pritsker [Pri08] by considering the integral over the product of unit disks instead of the unit torus as in the standard case. As in…

In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Amp\`ere type equations subject…

All continuous, SL(n) covariant valuations on Orlicz spaces are completely classified without any symmetric assumptions. It is shown that the moment matrix is the only such valuation if n\geq3, while a new functional shows up in dimension…

We give many examples of surfaces of general type with $p_g=0$ for which Bloch's conjecture holds, for all values of $K^2$ except 9. Our surfaces are equipped with an involution.

The problem of $L^p(R^3)\to L^2(S)$ Fourier restriction estimates for smooth hypersurfaces S of finite type in R^3 is by now very well understood for a large class of hypersurfaces, including all analytic ones. In this article, we take up…

We begin by studying the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of the semi-axes. We write down an explicit formula as an integral over the unit sphere in n-dimensions and use this…

We define a new type of valuation of a ring that combines the notion of Krull valuation with that of multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations. This…

We show an analogue of the Klain-Schneider theorem for valuations that are invariant under rotations around a fixed axis, called zonal. Using this, we establish a new integral representation of zonal valuations involving mixed area measures…

The moduli space of twisted holomorphic 1-forms on Riemann surfaces, equivalently dilation surfaces with scaling, admits a stratification and GL(2,R)-action as in the case of moduli spaces of translation surfaces. We produce an analogue of…

We show that the hyperplane conjecture holds for the classes of $k$-intersection bodies with arbitrary measures in place of volume.

For a measure space $(\Omega, \Sigma, \mu)$ with a positive finite measure $\mu$, and a positive real number $p$, we define the space $L_p^{+}(\mu)=L_p^{+}$ of all (equivalence classes of) $\Sigma$-measurable complex functions $f$ defined…

We obtain matching direct and inverse theorems for the degree of weighted $L_p$-approximation by polynomials with the Jacobi weights $(1-x)^\alpha (1+x)^\beta$. Combined, the estimates yield a constructive characterization of various…

Here one will find a rigorous treatment of the simplest situation in Surface Area Theory, viz. the nonparametric case with domain the unit square in the plane. This is installment IV of a four part discussion of certain aspects of Real…

We prove certain $L^p$ estimates ($1<p<\infty$) for non-isotropic singular integrals along surfaces of revolution. As an application we obtain $L^p$ boundedness of the singular integrals under a sharp size condition on their kernels.

For a certain class of varieties X, we derive a formula for the valuation d_{X} on the arc space L(Y) of a smooth ambient space Y, in terms of an embedded resolution of singularities. A simple transformation rule yields a formula for the…

Pseudo-cones are a class of unbounded closed convex sets, not containing the origin. They admit a kind of polarity, called copolarity. With this, they can be considered as a counterpart to convex bodies containing the origin in the…

We prove several variations on the results of Ricci and Travaglini concerning bounds for convolution with all rotations of a measure supported by a fixed convex curve in the plane. Estimates are obtained for averages over higher-dimensional…