Lutwak-Petty projection inequalities for Minkowski valuations and their duals

Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for intersection bodies of all orders by Leng and Lu is proved. These results are related to Grinberg's inequalities for dual affine quermassintegrals.Comment: 27 page

Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces

We consider flags $E_\bullet=\{X\supset E\supset \{q\}\}$, where $E$ is an exceptional divisor defining a non-positive at infinity divisorial valuation $\nu_E$ of a Hirzebruch surface $\mathbb{F}_\delta$ and $X$ the surface given by $\nu_E,$ and determine an analogue of the Seshadri constant for pairs $(\nu_E,D)$, $D$ being a big divisor on $\mathbb{F}_\delta$. The main result is an explicit computation of the vertices of the Newton-Okounkov bodies of pairs $(E_\bullet,D)$ as above, showing that they are quadrilaterals or triangles and distinguishing one case from another

Newton-Okounkov bodies of exceptional curve valuations

We prove that the Newton-Okounkov body of the flag $E_{\bullet}:= \left\{ X=X_r \supset E_r \supset \{q\} \right\}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which turn out to be triangular.Comment: 30 pages, 11 figures. V2: The terminology "infinitely singular valuations" used in the first version has been just replaced with the more customary "exceptional curve valuations" (it is simply a matter of denomination

Volume of line bundles via valuation vectors (different from Okounkov bodies)

Up to a factor 1/n!, the volume of a big line bundle agrees with the Euclidean volume of its Okounkov body. The latter is the convex hull of top rank valuation vectors of sections, all with respect to a single flag. In this text we give a different volume formula, valid in the ample cone, also based on top rank valuation vectors, but mixing data along several different flags

GL(n) equivariant Minkowski valuations

A classification of all continuous GL(n) equivariant Minkowski valuations on convex bodies in $\mathbb{R}^n$ is established. Together with recent results of F.E. Schuster and the author, this article therefore completes the description of all continuous GL(n) intertwining Minkowski valuations.Comment: 13 page

SL(n)-Contravariant LpL_p-Minkowski Valuations

All SL(n)-contravariant $L_p$-Minkowski valuations on polytopes are completely classified. The prototypes of such valuations turn out to be the asymmetric $L_p$-projection body operators.Comment: see arXiv:1209.3980 for related result

Integral geometric formulae for Minkowski tensors

The Minkowski tensors are the natural tensor-valued generalizations of the intrinsic volumes of convex bodies. We prove two complete sets of integral geometric formulae, so called kinematic and Crofton formulae, for these Minkowski tensors. These formulae express the integral mean of the Minkowski tensors of the intersection of a given convex body with a second geometric object (another convex body in the kinematic case and an affine subspace in the Crofton case) which is uniformly moved by a proper rigid motion, in terms of linear combinations of the Minkowski tensors of the given geometric objects

Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any semigroup in the lattice Z^n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results: we show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of Fujita approximation theorem for semigroups of integral points and graded algebras, which imply a generalization of this theorem for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of the Kushnirenko theorem (from Newton polytope theory) and a new version of the Hodge inequality. We also give elementary proofs of the Alexandrov-Fenchel inequality in convex geometry and its analogue in algebraic geometry.Comment: 39 pages. Revised in several places and the title slightly modified. Final version, to appear in Annals of Mathematic

Convex bodies and algebraic equations on affine varieties

Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on degree of subvarieties of a projective space (in algebraic geometry) and Brunn-Minkowski inequality (in convex geometric).Comment: Preliminary version, may contain several typos, 44 page

SL(n)-Covariant LpL_p-Minkowski Valuations

All continuous SL(n)-covariant $L_p$-Minkowski valuations defined on convex bodies are completely classified. The $L_p$-moment body operators turn out to be the nontrivial prototypes of such maps.Comment: see arXiv:1410.7021 for related result