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Fernanda M. Baêta

Publications and source records attributed to Fernanda M. Baêta.

6 recordsLinked to original sources
arXiv2026

A Characterization of Functional Affine Surface Areas

A characterization of valuations on the space of convex Lipschitz functions whose domain is a polytope in $\mathbb{R}^n$ is obtained. It is shown that every upper semicontinuous, equi-affine and dually epi-translation invariant valuation can be written as a linear combination of a constant term, the volume of the domain, and a functional affine surface area. In addition, dual statements for finite-valued convex functions are established.

math.MG↗
arXiv2026

Affine chord Sobolev inequalities and radial mean bodies for functions

Affine isoperimetric inequalities for the functional radial mean bodies are derived from the new affine chord Sobolev inequalities, which extend the recent affine isoperimetric inequalities of Haddad and Ludwig from convex bodies to functions. The affine chord Sobolev inequalities further imply a strengthening of the Euclidean chord Sobolev inequalities introduced by Baêta and Cai. Moreover, for $s$-concave functions $f$ with compact support and $s>0$, a parameter-dependent monotonicity property of the functional radial mean body $R_α f$ is obtained: $R_αf \subset R_βf$ for $-1 < α< β$, and, after suitable normalization, the reverse inclusion also holds. These sharp results generalize the corresponding monotonicity for geometric radial mean bodies established by Gardner and Zhang.

math.MG↗
arXiv2026

Chord Sobolev inequalities

The paper establishes a new family of sharp analytic inequalities. Together with the fractional Sobolev inequalities of Almgren and Lieb, they form a complete class of analytic inequalities, referred to as the chord Sobolev inequalities. A close connection between these inequalities and chord isoperimetric inequalities in integral geometry is established through a functional extension of chord power integrals. The limiting cases of the chord Sobolev inequalities are derived, one of which yields a logarithmic Sobolev-type inequality. Combined with the work of Bourgain, Brezis, and Mironescu, these results complete the picture of the chord Sobolev inequalities, including their endpoint cases.

math.MG↗
arXiv2025

On the Semicontinuity of Functionals on Function Spaces

Results on the upper and lower semicontinuity of functionals defined on spaces of convex and more general functions are established. In particular, the following result is obtained. Let $ϕ(v; \cdot)$ be the density of the absolutely continuous part of a Radon measure $Φ(v; \cdot)$ associated to a function $v\colon X\rightarrow \mathbb{R}$ defined on the topological measure space $(X,λ)$. For concave $ζ\colon [0, \infty)\rightarrow[0,\infty)$ with $\lim_{t\to 0} ζ(t)=0$ and $\lim_{t\to\infty}ζ(t)/t= 0$, it is shown that the functional $v \mapsto \int_{X} ζ(ϕ(v;x))dλ(x)$ depends upper semicontinuously on $v$. Examples include functional affine surface areas for convex functions.

math.FA↗
arXiv2025

Asymptotic Weighted Approximation of Convex Functions

Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to infinity. These asymptotic expressions are formulated in terms of a functional that extends the notion of affine surface area to the functional setting.

math.OC↗