Related papers: Orlicz Mixed Affine Quermassintegrals
We interpret the log-Brunn-Minkowski conjecture of B\"or\"oczky-Lutwak-Yang-Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert-Brunn-Minkowski operator coincides with the…
The Brunn-Minkowski theory relies heavily on the notion of mixed volumes. Despite its particular importance, even explicit representations for the mixed volumes of two convex bodies in Euclidean space are available only in special cases.…
In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of…
The variation of a class of Orlicz moments with respect to the Asplund sum within the class of log-concave functions is demonstrated. Such a variational formula naturally leads to a family of dual Orlicz curvature measures for log-concave…
The author established the affine Orlicz Polya-Szego principle for log-concave functions and conjectured that the principle can be extended to the general Orlicz Sobolev functions. In this paper, we confirm this conjecture completely. An…
A sharp quantitative version of the $L_p-$mixed volume inequality is established. This is achieved by exploiting an improved Jensen inequality. This inequality is a generalization of Pinsker-Csisz\'ar-Kullback inequality for the Tsallis…
We introduce f-divergence, a concept from information theory and statistics, for convex bodies in R^n. We prove that f-divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We…
In a seminal paper "Volumen und Oberfl\"ache" (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the…
In this paper we study the dual Orlicz-Minkowski problem, which is a generalization of the dual Minkowski problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial…
In an earlier paper \cite{mazeng} the authors introduced the notion of curvature entropy, and proved the plane log-Minkowski inequality of curvature entropy under the symmetry assumption. In this paper we demonstrate the plane log-Minkowski…
The Orlicz-Legendre ellipsoids, which are in the framework of emerging dual Orlicz Brunn-Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz-John ellipsoids, and have largely…
An extension of Marcinkiewicz Interpolation Theorem, allowing intermediate spaces of Orlicz type, is proved. This generalization yields a necessary and sufficient condition so that every quasilinear operator, which maps the set, $S(X,\mu)$,…
Three new combinations of convex bodies are introduced and studied: the $L_p$ fiber, $L_p$ chord and graph combinations. These combinations are defined in terms of the fibers and graphs of pairs of convex bodies, and each operation…
We prove an analogue of the classical Steiner formula for the $L_p$ affine surface area of a Minkowski outer parallel body for any real parameters $p$. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual…
We provide general estimates which compare the quermassintegrals of a convex body $K$ in ${\mathbb R}^n$ with the averages of the corresponding quermassintegrals of the $k$-codimensional sections of $K$ over $G_{n,n-k}$. An example is the…
An affine version of the linear subspace concentration inequality as proposed by Wu is established for centered convex bodies. This generalizes results from Wu and Freyer, Henk, Kipp on polytopes to convex bodies.
In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we discussed how recent developments on concavity of measures have laid the foundations of a nascent weighted Brunn-Minkowski theory. In particular, we defined the mixed…
Building on work of Furstenberg and Tzkoni, we introduce ${\bf r}$-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the…
In this paper, we provide an affirmative answer to [16, Conjecture 1.5] on the Alexandrov-Fenchel inequality for quermassintegrals for convex capillary hypersurfaces in the Euclidean half-space. More generally, we establish a theory for…
Let $C$ be a closed convex cone in ${\mathbb R}^n$, pointed and with interior points. We consider sets of the form $A=C\setminus A^\bullet$, where $A^\bullet\subset C$ is a closed convex set. If $A$ has finite volume (Lebesgue measure),…