arXiv subjects

Auttawich Manui

Publications and source records attributed to Auttawich Manui.

7 recordsLinked to original sources

arXiv2026

Volume and Projection Inequalities II: Determinants and $L_p$-Sums

We study inequalities for the volume of orthogonal projections and their relation to Firey $L_p$-sum, together with their determinant-power analogues, motivated by the Dembo--Cover--Thomas conjecture. For $L_p$-zonoids $K,L\subset\mathbb{R}^n$ and $u\in S^{n-1}$, we consider the inequality \[ \left( \frac{|K\oplus_p L|} {|P_{u^\perp}(K\oplus_p L)|} \right)^p \geq \left( \frac{|K|}{|P_{u^\perp}K|} \right)^p + \left( \frac{|L|}{|P_{u^\perp}L|} \right)^p . \] For every $1<p<2$, we prove that this inequality fails in every dimension $n\geq2$. In contrast, the weak one-term inequality, obtained by omitting the second term on the right-hand side, holds in dimension two throughout the full range $1\leq p\leq2$. The proof of this planar result uses a sharp estimate for the normalized duality map. We also classify the corresponding determinant-power inequalities in the range $0<p<2$. The strong two-term inequality holds in dimension two and fails in every dimension $n\geq3$. The weak one-term inequality holds for $0<p\leq1$ in dimensions $n\leq3$ and fails for $n\geq4$; for $1<p<2$, it holds only in dimension two.

math.MG↗

arXiv2026

Volume and Projection Inequalities I: Zonoids and Courtade's Conjecture

We study volume and projection inequalities for zonoids through the multiaffine determinant polynomials that encode their volumes. We show that a log-submodularity conjecture for the volume of zonoids is equivalent to the Rayleigh property of zonotope volume polynomials, which we prove for the case when the degree or codegree is at most 3. This result is sharp in that when the degree and codegree are at least 4, we construct counterexamples using the existence of non-Rayleigh matroids in ranks at least four. Additionally, we provide unimodular or graphical counterexamples in dimensions four and higher to an equivalent projection inequality formulation of the conjecture. We also show that a stronger projection inequality fails already in dimension three. We next disprove Courtade's conjecture using a pair of orthogonal double bodies of revolution. Although Courtade's conjecture was originally formulated for general convex bodies, we show that it fails even for zonoids in every dimension at least three.

math.MG↗

arXiv2026

Integral inequalities for $\alpha$-convolutions of $\alpha$-concave functions

Classical sumset inequalities originating in additive combinatorics admit geometric analogues for convex bodies in finite-dimensional real vector spaces, as recently developed by Fradelizi and two of the authors. We develop integral analogues for geometric $\alpha$-concave functions under $\alpha$-convolutions, an operation that arises naturally in the ``geometrization of probability'' program. In particular, we establish a Pl\"unnecke--Ruzsa-type inequality, as well as sharp analogues of sum-difference and Ruzsa triangle inequalities, for $\alpha$-convolutions of $\alpha$-concave functions. We also prove a sharp Rogers--Shephard-type inequality and characterize its equality cases for $\alpha$-concave functions, bridging the log-concave case studied by Alonso-Guti\'errez, Gonz\'alez-Merino, Jim\'enez, and Villa and the quasi-concave case studied by Colesanti.

math.FA↗

arXiv2026

L_p-Rogers--Shephard type inequalities for L_p-zonoids and symmetric bodies

We study generalizations of the classical Rogers--Shephard inequalities in the framework of Firey $L_p$-summation. We first consider the class of asymmetric $L_p$-zonoids. In this setting, we show that proving a sharp $L_p$-Rogers--Shephard inequality for asymmetric $L_p$-zonoids in $\mathbb{R}^n$ is equivalent to proving a sharp inequality between the volumes of projections of $B_q^m\cap \mathbb{R}^m_+$ and $B_q^m$ onto an $n$-dimensional subspace $E$, where $q$ is the H\"older conjugate of $p$. We conjecture that the inequality is sharp when the subspace $E$ is a coordinate subspace. We fully establish this inequality along with equality conditions in the case $p =2$. For general $p$, we prove it in the case $n=m-1$, $n=1$, and discuss several particular cases, including an averaged version and a local version of the inequality. We then turn to the setting of convex bodies having a center of symmetry. Rogers and Shephard also proved a sharp version of their inequality for bodies in this class. We conjecture a similar bound for the $L_p$-summation, and we establish our conjecture for the particular case of asymmetric $L_1$-zonoids, which, in particular, proves our conjecture in the planar case.

math.MG↗

arXiv2026

Equality cases for the $L_p$-Rogers--Shephard inequality in the plane and for locally anti-blocking bodies in $\mathbb{R}^n$

The classical Rogers--Shephard inequalities were extended to the Firey $L_p$-summation by Bianchini and Colesanti in the plane and by Zvavitch and the second and fourth authors for locally anti-blocking convex bodies in $\mathbb{R}^n$, leaving open the equality cases. We characterize the equality cases of these inequalities: in both cases, for $p>1$, equality holds if and only if the convex body is a simplex with one vertex at the origin.

math.MG↗

arXiv2026

On the Fourier Mean Bodies of a Convex Body

In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_pK$ of a convex body $K\subset\mathbb R^n$, $p>-1$, which have since become important objects in geometric tomography. In this paper we study the Fourier transforms of the radial functions of $R_pK$. This leads to a new family of star-shaped sets $F_pK$, which we call the Fourier $p$th mean bodies of $K$. We prove Fourier inversion formulas connecting $R_pK$ and $F_pK$, realizing them as $p$-intersection bodies in the sense of A. Koldobsky. We develop the basic affine geometry of $F_pK$; this includes affine invariance and monotonicity properties. We identify the range of $p$ where $F_p K$ is compact in terms of the decay of $|\widehat{\chi_K}|^2$. We show that $F_pK$ is an origin-symmetric convex body for every $0<p\le1$. This range is sharp in general: already for the cube, $F_p[-1,1]^n$ is not convex for $1<p<2$ and $n\geq 2,$ while $F_p[-1,1]^n$ is not compact for $p\geq 2$. We further investigate the features Fourier mean bodies share with intersection bodies: we prove Hensley-type estimates for $F_pK$ when $K$ is isotropic and investigate a few affine isoperimetric inequalities.

math.MG↗

arXiv2024

On the volume of sums of anti-blocking bodies

We study inequalities on the volume of Minkowski sum in the class of anti-blocking bodies. We prove analogues of Pl\"unnecke-Ruzsa type inequality and V. Milman inequality on the concavity of the ratio of volumes of bodies and their projections. We also study Firey $L_p-$sums of anti-blocking bodies and prove Pl\"unnecke-Ruzsa type inequality; V. Milman inequality and Roger-Shephard inequality. The sharp constants are provided in all of those inequalities, for the class of anti-blocking bodies. Finally, we extend our results to the case of unconditional product measures with decreasing density.

math.MG↗