arXiv subjects

Mohamed A. Mouamine

Publications and source records attributed to Mohamed A. Mouamine.

5 recordsLinked to original sources

arXiv2026

Higher-order Godbersen inequality and a characterization of simplices via the face lattice

We settle the long-standing Godbersen conjecture that the mixed volume of a convex body of fixed positive volume with its reflection about the origin is maximized precisely by simplices. For the proof, we establish a new characterization of simplices by a natural incidence relation on the face lattice. Moreover, we establish a higher-order generalization of the Godbersen inequality and show that simplices are the only nontrivial equality cases, confirming a conjecture of Schneider from 2000.

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arXiv2026

Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality

We prove that the mixed volume of a convex body of fixed positive volume with its reflection about the origin is maximized by simplices. This confirms a conjecture of C. Godbersen from 1938 and refines the classical Rogers-Shephard inequality. We also prove that simplices are the only extremizers among convex polytopes. Finally, we use this inequality to prove an $L_p$-version of the Rogers-Shephard inequality for convex bodies containing the origin and show that, for any $p\in(1,\infty]$, the only extremizers are simplices with a vertex at the origin.

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arXiv2026

Additive Kinematic Formulas for Functional Minkowski Vectors

We establish an additive kinematic formula for the functional Minkowski vectors using mixed Monge-Ampère measures. These vectors, recently introduced and characterized by the author and F. Mussnig, form a natural family of vector-valued valuations on the space of convex functions. This result represents the first integral geometric application of this characterization.

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arXiv2025

The Vectorial Hadwiger Theorem on Convex Functions

A complete classification of continuous, dually epi-translation invariant, and rotation equivariant valuations on convex functions is established. This characterizes the recently introduced functional Minkowski vectors, which naturally extend the classical Minkowski relations. For this, the existence of these operators with singular densities is shown, along with additional representations involving mixed Monge-Ampère measures, Kubota-type formulas, and area measures of higher dimensional convex bodies. Dual results are formulated for valuations on super-coercive convex functions.

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arXiv2025

A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions

A functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions is established, providing a classification of continuous, translation covariant, simple valuations. Under additional rotation equivariance assumptions, an analytic counterpart of the moment vector is characterized alongside a new epi-translation invariant valuation. The former arises as the top-degree operator in a family of functional intrinsic moments, which are linked to functional intrinsic volumes through translations. The latter represents the top-degree operator in a class of Minkowski vectors, which are introduced in this article and which lack classical counterparts on convex bodies, as they vanish due to the Minkowski relations. Additional classification results are obtained for homogeneous valuations of extremal degrees.

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