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Jacopo Ulivelli

Publications and source records attributed to Jacopo Ulivelli.

14 recordsLinked to original sources
arXiv2026

A strengthening of the dimensional Brunn-Minkowski conjecture implies the (B)-conjecture

We prove that if a sufficiently regular even log-concave measure satisfies a certain stronger form of the dimensional Brunn-Minkowski conjecture, then it also satisfies the (B)-conjecture. Furthermore, we show that hereditarily convex measures satisfy the aforementioned strengthened form, therefore providing an alternative proof of a recent result by Cordero-Erausquin and Eskenazis stating that a hereditarily convex measure satisfies both conjectures.

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arXiv2026

A new infinitesimal form of the Pr\'ekopa-Leindler inequality with multiplicative structure and applications

By differentiating a concavity principle arising from the Pr\'ekopa-Leindler inequality, we obtain a statement simultaneously strengthening the weighted boundary Poincar\'e inequality and the Brascamp-Lieb variance inequality. The resulting inequality possesses a multiplicative structure, which we exploit to develop an alternative to the (by now classical) $L_2$ method in the study of geometric and analytic inequalities. We apply this approach to derive a stability estimate for the weighted Poincar\'e inequality and to investigate the dimensional Brunn-Minkowski conjecture. In particular, in the latter setting, we obtain new reformulations together with several partial results.

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arXiv2025

Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries

An explicit solution to the Christoffel-Minkowski problem for convex bodies of revolution is presented. The conditions on the prescribed measure involve only first moments over spherical caps, and the support function of the resulting convex body is given by an explicit representation formula in terms of the measure. More generally, existence problems for mixed area measures are addressed. The approach relies on constructing explicit convex solutions to mixed Monge-Amp\`ere equations on $\mathbb{R}^n$ under the assumption of radial symmetry, with the conditions on the measure being expressed through its values on open balls. As a special case, the Dirichlet problem for $k$-Hessian equations on $\mathbb{R}^n$ is treated.

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arXiv2024

Inequalities and counterexamples for functional intrinsic volumes and beyond

We show that analytic analogs of Brunn-Minkowski-type inequalities fail for functional intrinsic volumes on convex functions. This is demonstrated both through counterexamples and by connecting the problem to results of Colesanti, Hug, and Saor\'in G\'omez. By restricting to a smaller set of admissible functions, we then introduce a family of variational functionals and establish Wulff-type inequalities for these quantities. In addition, we derive inequalities for the corresponding family of mixed functionals, thereby generalizing an earlier Alexandrov-Fenchel-type inequality by Klartag and recovering a special case of a recent P\'olya-Szeg\H{o}-type inequality by Bianchi, Cianchi, and Gronchi.

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arXiv2024

Polynomial valuations on convex functions and their maximal extensions

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the support of these valuations. The results rely on a homogeneous decomposition for the space of polynomial valuations of bounded degree and the support properties of certain distributions associated to the homogeneous components. As an application, an explicit integral representation for valuations of top degree is established.

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arXiv2024

Additive kinematic formulas for convex functions

We prove a functional version of the additive kinematic formula as an application of the Hadwiger theorem on convex functions together with a Kubota-type formula for mixed Monge-Amp\`ere measures. As an application, we give a new explanation for the equivalence of the representations of functional intrinsic volumes as singular Hessian valuations and as integrals with respect to mixed Monge-Amp\`ere measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on $\operatorname{SO}(n-1)\times \operatorname{O}(1)$ is considered.

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arXiv2024

Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions

The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric-type inequalities for convex bodies. Following a classical work by Schneider, both inequalities have been extended to the so-called $m$th-order setting. In this work, we establish the $m$th-order analogues for these inequalities in the setting of log-concave functions. Our proof of the functional Zhang's projection inequality employs properties of the asymmetric LYZ body, significantly streamlining the argument and producing a novel approach for the case $m=1$. Furthermore, we introduce and analyze the radial mean bodies of a log-concave function, thereby providing a functional generalization of Gardner and Zhang's radial mean bodies. These are new even in the case $m=1$. Our development leverages an extension of Ball bodies, which may be of independent interest.

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arXiv2024

Kubota-type formulas and supports of mixed measures

Kubota's integral formula expresses the intrinsic volumes of a convex body as averages over its projections onto linear subspaces. In this work, we introduce a new class of Kubota-type formulas for mixed area measures adapted to rotations around a fixed axis, which encode a crucial disintegration property. Our construction is motivated by applications to valuations on convex functions. In the latter framework, we obtain corresponding statements for (conjugate) mixed Monge-Amp\`ere measures. As a by-product, we characterize supports of mixed area and mixed Monge-Amp\`ere measures, thereby confirming a special case of a conjecture by Schneider.

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arXiv2023

First variation of functional Wulff shapes

We introduce functional Wulff shapes based on the classical construction for compact convex sets. With this new tool, we establish a functional version of Aleksandrov's variational lemma in the family of convex functions with compact domain. The resulting formula is then applied to evaluate the first variation of a class of functionals on convex functions. In particular, we extend a recent result by Huang, Liu, Xi, and Zhao.

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arXiv2023

From valuations on convex bodies to convex functions

A geometric framework relating valuations on convex bodies to valuations on convex functions is introduced. It is shown that a classical result by McMullen can be used to obtain a characterization of continuous, epi-translation invariant, and n-epi-homogeneous valuations on convex functions, which was previously established by Colesanti, Ludwig, and Mussnig. Following an approach by Goodey and Weil, a new characterization of 1-epi-homogeneous valuations is obtained.

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arXiv2023

The (Self-Similar, Variational) Rolling Stones

The interplay between variational functionals and the Brunn-Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from B\"or\"oczky, Lutwak, Yang, and Zhang. We show how these results imply the existence of self-similar solutions to variational flow problems \`a la Firey's worn stone problem.

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arXiv2022

Convergence properties of symmetrization processes

Steiner symmetrization is well known for its rounding and general convergence properties. We identify a whole family of symmetrizations sharing analogue behaviors: In fact we prove that all these symmetrizations share the same converging symmetrization processes, together with some pathological phenomena.

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arXiv2021

Generalization of Klain's Theorem to Minkowski Symmetrization of compact sets and related topics

We shall prove a convergence result relative to sequences of Minkowski symmetrals of general compact sets. In particular, we investigate the case when this process is induced by sequences of subspaces whose elements belong to a finite family, following the path marked by Klain in [13], and the generalizations in [4] and [2]. We prove an analogue result for Fiber symmetrization of a specific class of compact sets. The idempotency for symmetrization of this family of sets is investigated, leading to a simple generalization of a result from Klartag [14] regarding the approximation of a ball through a finite number of symmetrizations, and generalizing an approximation result in [9]

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