Removing the finite-perimeter hypothesis from the Figalli-Jerison convexity-from-marginals theorem

The Figalli-Jerison theorem (Journal of Functional Analysis, 2014) states that a bounded set of finite perimeter in $\mathbb{R}^n$, $n \geq 2$, whose marginals onto hyperplanes are log-concave for almost every direction, is convex up to a set of measure zero — a converse to the Brunn-Minkowski marginal concavity of convex sets. The finite-perimeter hypothesis is consumed twice in the proof (an a priori bound on a quadratic Hilbertian $H^{1/2}$ energy of a Gaussian smoothing, and a localized perimeter lower bound at reduced-boundary points), and the authors state that the hypothesis is technical and that they expect the result to be true without it. The problem: prove, or refute, that the theorem holds for arbitrary bounded measurable sets; and determine the minimal regularity actually needed. Follow-up literature resolved much of the companion analytic program on the underlying Hilbertian norm — Hernandez (2017) proved that the limsup of the scaled norm characterizes finite perimeter, computed the exact limit for finite-perimeter sets, and constructed infinite-perimeter sets with vanishing liminf (showing the original compactness scheme has no naive extension), while Poliakovsky (2020) established exact jump-set formulas for mollified $W^{1/q,q}$ norms of bounded BV functions — but the geometric statement is untouched: a 2024 survey of analytic characterizations of convex sets restates it recording only the authors' expectation. The problem is open.

Background

Let $E \subset \mathbb{R}^n$ be a Borel set and, for a unit direction $e \in S^{n-1}$, define its marginal onto the hyperplane $e^\perp = \{x : e \cdot x = 0\}$ by

$$w_e(x') = \int_{\mathbb{R}} 1_E(x' + te)\,dt, \qquad x' \in e^\perp,$$

where $1_E$ is the characteristic function. Thus $w_e(x')$ is the length of the one-dimensional slice of $E$ along the line through $x'$ parallel to $e$ — the continuous analogue of an X-ray of $E$ in direction $e$. A nonnegative function $w$ is called log-concave if $w = e^{-V}$ for some convex $V$ taking values in $\mathbb{R} \cup \{+\infty\}$. If $E$ is convex, the Brunn-Minkowski inequality (in its marginal form; see R. J. Gardner's survey) gives an even stronger property: $w_e^{1/(n-1)}$ is concave on $e^\perp$ for every $e$, so in particular $w_e$ is log-concave.

Figalli and Jerison asked for the converse: if a set has log-concave marginals in almost every direction (with respect to the surface measure $\mathcal{H}^{n-1}$ on the sphere), must it be convex? The question is meaningful for $n \geq 2$ (for $n = 1$ the only marginal is the constant $|E|$, so the hypothesis degenerates). The converse is not soft: for a smoothly bounded non-convex set, some directional marginal develops an infinite derivative at an interior point of its support, but this heuristic fails for irregular sets — Figalli and Jerison note the example of a disk with a small square removed from its interior. What they prove is the converse under a regularity hypothesis: $E$ is a set of finite perimeter, meaning the distributional gradient $\nabla 1_E$ is a finite Radon measure; its total mass $P(E)$ is the perimeter and equals $\mathcal{H}^{n-1}(\partial^* E)$, the measure of the reduced boundary $\partial^* E$ of De Giorgi's theory of sets of finite perimeter (the measure-theoretic boundary, on which $E$ has an approximate tangent plane; sets of finite perimeter are the level sets of BV functions, i.e. functions of bounded variation whose distributional derivative is a finite measure). Their Theorem 1.1: if $E$ is bounded, has finite perimeter, and for $\mathcal{H}^{n-1}$-a.e. $e$ the support $A_e = \{w_e > 0\}$ is convex, $w_e$ is locally Lipschitz inside $A_e$, and the Lipschitz constants on the $\delta$-interior $A_e^\delta = \{x \in A_e : \mathrm{dist}(x, \partial A_e) \geq \delta\}$ are bounded by $C_\delta$ uniformly in $e$, then $E$ is convex up to a set of measure zero (equivalently, its set of density points is convex). Since log-concave functions are automatically uniformly Lipschitz strictly inside their support, this yields Corollary 1.2: a bounded set of finite perimeter whose marginals are log-concave for a.e. $e$ is convex up to measure zero.

The proof measures perimeter through a quadratic Hilbertian functional. With $\varphi_\varepsilon = 1_E * \gamma_{n,\varepsilon}$ the convolution with a Gaussian of scale $\varepsilon$, and the Gagliardo semi-norm of order 1/2 defined by $\|u\|_{H^{1/2}}^2 = \iint \frac{|u(x)-u(y)|^2}{|x-y|^{n+1}}\,dx\,dy$, their Theorem 1.3 shows $\frac{1}{|\log \varepsilon|}\|\varphi_\varepsilon\|_{H^{1/2}}^2 \simeq P(E)$ as $\varepsilon \to 0$ (comparable up to dimensional constants), together with a localized version. In dimension $n = 2$ the Fourier identity $\|\varphi\|_{H^{1/2}(\mathbb{R}^2)}^2 \simeq \int_0^\pi \int_{\mathbb{R}} |w_\theta'|^2\,dt\,d\theta$ converts this energy into marginal information; the marginal regularity forces the blow-up of the energy measures to concentrate on the boundary of the projection of the convex hull of $E$, contradicting the localized lower bound at a reduced-boundary point of $E$ lying inside that hull. Finite perimeter is used in both the upper bound (a BV estimate $\|\varphi_\varepsilon\|_{H^{1/2}}^2 \leq C|\log \varepsilon|\, P(E)$, Lemma 2.1) and the lower bound at reduced-boundary points (Lemma 2.2).

The authors explicitly flag the regularity hypothesis as an artifact: they write that the assumption of finite perimeter is technical and that they expect the result to be true without it, while finite perimeter plays an important role in the proof, which is based on measuring the perimeter using a fractional Sobolev norm of a smoothing of $1_E$. They also record a second uninvestigated question about this analytic engine: the connection, if any, between their Gaussian-mollification quadratic $H^{1/2}$ functional and the $s$-fractional perimeters of Caffarelli-Roquejoffre-Savin (the perimeters $P_s(E) = \int_E \int_{E^c} |x-y|^{-(n+2s)}\,dx\,dy$ of nonlocal minimal surface theory), and their Gamma-convergence to the classical perimeter as established by Ambrosio-De Philippis-Martinazzi and Caffarelli-Valdinoci — noting that because their norm is quadratic (Hilbertian) rather than non-quadratic, the analysis of those works does not transfer.

Subsequent work developed this analytic thread but not the geometric statement. Hernandez (2017, preprint) proved that the $\limsup$ of the scaled functional characterizes finite perimeter (it is infinite exactly when $P(E) = \infty$), that the exact limit $\lim_{\varepsilon \to 0} \frac{1}{|\log \varepsilon|}\|\gamma_\varepsilon * 1_E\|_{H^{1/2}}^2 = c_n P(E)$ exists for finite-perimeter sets, and — crucially for the geometric question — constructed bounded sets of infinite perimeter along which the $\liminf$ vanishes along a sequence of scales, so the compactness scheme of the Figalli-Jerison proof has no naive extension to general bounded sets. Poliakovsky (2020), motivated by both papers, established the exact jump-set formula $\lim_{\varepsilon \to 0} \frac{1}{|\ln \varepsilon|}\|\eta_\varepsilon * u\|_{W^{1/q,q}}^q = C_0 \int_{J_u} |u^+ - u^-|^q\,d\mathcal{H}^{N-1}$ for all $u \in BV \cap L^\infty$ and $q > 1$ ($J_u$ the jump set, $u^\pm$ the one-sided traces), placing the quadratic functional inside the Bourgain-Brezis-Mironescu (BBM) family of nonlocal functionals that recover total variation under logarithmic scaling. The Figalli-Jerison Lemma 2.1 also circulates as a tool in PDE work on stable solutions (Figalli-Serra; Brezis-Van Schaftingen-Yung). None of this addresses whether log-concavity of the marginals forces convexity of a set whose perimeter is not assumed finite; that is the open problem below.

Problem Statement

Remove the finite-perimeter hypothesis from the Figalli-Jerison convexity-from-marginals theorem. For $n \geq 2$, prove or disprove the following statement (the source paper's Theorem 1.1 with its regularity hypothesis deleted):

Let $E \subset \mathbb{R}^n$ be a bounded measurable set. For a unit direction $e \in S^{n-1}$ (the unit sphere of $\mathbb{R}^n$) write $e^\perp = \{x \in \mathbb{R}^n : e \cdot x = 0\}$ for the hyperplane orthogonal to $e$, and let $w_e(x') = \int_{\mathbb{R}} 1_E(x' + te)\,dt$ (with $1_E$ the characteristic function of $E$) be the marginal of $E$ onto $e^\perp$, i.e. the length of the slice of $E$ along the line through $x'$ parallel to $e$; set $A_e = \{x' \in e^\perp : w_e(x') > 0\}$. Assume that for $\mathcal{H}^{n-1}$-a.e. $e \in S^{n-1}$ (i.e. for every direction $e$ outside a set of vanishing $(n-1)$-dimensional Hausdorff measure on the sphere): (i) $A_e$ is convex; and (ii) $w_e$ is locally Lipschitz inside $A_e$, and for every $\delta > 0$ there is a constant $C_\delta$, independent of $e$, such that $|\nabla w_e| \leq C_\delta$ a.e. on the set of points of $A_e$ at distance at least $\delta$ from $\partial A_e$. (Since log-concave functions have convex support and are uniformly Lipschitz strictly inside their support, these hypotheses are satisfied in particular when $w_e$ is log-concave, i.e. $w_e = e^{-V_e}$ for some convex $V_e : e^\perp \to \mathbb{R} \cup \{+\infty\}$, for a.e. $e$; that special case is the source paper's Corollary 1.2.)

Then must $E$ be convex up to a set of measure zero, i.e. must its set of density points $E^{(1)} = \{x \in \mathbb{R}^n : \lim_{r \to 0} |E \cap B_r(x)|/|B_r(x)| = 1\}$ (with $B_r(x)$ the ball of radius $r$ centered at $x$) be convex? The authors state the expectation that the answer is affirmative. An affirmative answer must dispose of the a priori finite-perimeter assumption entirely — no use of $P(E) < \infty$ (finiteness of the distributional gradient $\nabla 1_E$), of reduced-boundary structure, or of equiboundedness of the mollified $H^{1/2}$ energies (Hernandez constructed bounded sets of infinite perimeter for which the scaled energy has vanishing liminf, so these cannot hold in general). A negative answer is an explicit bounded measurable set, not convex up to measure zero, that satisfies hypotheses (i)-(ii) — in particular, one whose marginals are log-concave for a.e. direction.

The second component is the regularity threshold: determine what minimal regularity of $E$, if any beyond boundedness and measurability, is needed for the conclusion — that is, characterize the largest class of bounded sets for which log-concavity of the a.e. marginals forces convexity, with counterexamples showing the characterization is sharp. The source paper records only the expectation that no regularity at all is needed; the minimal-regularity formulation is the natural sharpening of that expectation, and a complete answer to the first component discharges it — an affirmative answer settles the minimal-regularity component, and a refutation of the statement settles the universal claim and resolves the task.

References

1. Alessio Figalli and David Jerison, "How to recognize convexity of a set from its marginals", Journal of Functional Analysis 266(3):1685-1701, 2014. DOI: 10.1016/j.jfa.2013.05.040; https://doi.org/10.1016/j.jfa.2013.05.040 (arXiv:1502.06511, https://arxiv.org/abs/1502.06511)
2. Felipe Hernandez, "Properties of a Hilbertian Norm for Perimeter", preprint, 2017. arXiv:1709.08262, https://arxiv.org/abs/1709.08262
3. Arkady Poliakovsky, "Asymptotic behavior of the W^{1/q,q}-norm of mollified BV functions and applications to singular perturbation problems", ESAIM: Control, Optimisation and Calculus of Variations 26, article 77, 2020. DOI: 10.1051/cocv/2019051; https://doi.org/10.1051/cocv/2019051 (arXiv:1812.06358, https://arxiv.org/abs/1812.06358)
4. Nikolay Kuznetsov, "Analytic Characterization of Convex Sets in ℝm (a Survey)", Journal of Mathematical Sciences 286(2):238-247, 2024. DOI: 10.1007/s10958-024-07501-6; https://doi.org/10.1007/s10958-024-07501-6 (arXiv:2405.18013, https://arxiv.org/abs/2405.18013; the arXiv version carries the slightly different title "On analytic characterization of convex sets in $\mathbb{R}^m$ (a survey)")
5. L. Caffarelli, J.-M. Roquejoffre and O. Savin, "Nonlocal minimal surfaces", Communications on Pure and Applied Mathematics 63(9):1111-1144, 2010. DOI: 10.1002/cpa.20331; https://doi.org/10.1002/cpa.20331
6. Luigi Ambrosio, Guido De Philippis and Luca Martinazzi, "Gamma-convergence of nonlocal perimeter functionals", Manuscripta Mathematica 134(3-4):377-403, 2011. DOI: 10.1007/s00229-010-0399-4; https://doi.org/10.1007/s00229-010-0399-4
7. Luis Caffarelli and Enrico Valdinoci, "Uniform estimates and limiting arguments for nonlocal minimal surfaces", Calculus of Variations and Partial Differential Equations 41(1-2):203-240, 2011. DOI: 10.1007/s00526-010-0359-6; https://doi.org/10.1007/s00526-010-0359-6
8. R. J. Gardner, "The Brunn-Minkowski inequality", Bulletin of the American Mathematical Society 39(3):355-406, 2002. DOI: 10.1090/S0273-0979-02-00941-2; https://doi.org/10.1090/S0273-0979-02-00941-2
9. Alessio Figalli and Joaquim Serra, "On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4 + 1", Inventiones mathematicae 219:153-177, 2020. DOI: 10.1007/s00222-019-00904-2; https://doi.org/10.1007/s00222-019-00904-2 (arXiv:1705.02781, https://arxiv.org/abs/1705.02781)
10. Haim Brezis, Jean Van Schaftingen and Po-Lam Yung, "Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail", Calculus of Variations and Partial Differential Equations 60(4):129, 2021. DOI: 10.1007/s00526-021-02001-w; https://doi.org/10.1007/s00526-021-02001-w (arXiv:2104.09867, https://arxiv.org/abs/2104.09867)
11. Dario Cordero-Erausquin and Alexandros Eskenazis, "Concavity principles for weighted marginals", preprint, 2025. arXiv:2506.16941, https://arxiv.org/abs/2506.16941