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We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke--Santal\'o inequality and the $L_p$-affine isoperimetric inequalities, for convex bodies in spherical space. Specifically, we establish an isoperimetric inequality for the floating area and prove a stability result based on the spherical volume difference. The floating area has previously been studied as a natural extension of classical affine surface area to non-Euclidean convex bodies in spaces of constant curvature. In this work, we introduce the $L_p$-floating areas for spherical convex bodies, extending Lutwak's centro-affine invariant family of $L_p$-affine surface area measures from Euclidean geometry. We prove a duality formula, monotonicity properties, and isoperimetric inequalities associated with this new family of curvature measures for spherical convex bodies. Additionally, we propose a novel curvature entropy functional for spherical convex bodies, based on the $L_p$-floating area, and establish a corresponding dual isoperimetric inequality. Finally, we extend our spherical notions to space forms with non-negative constant curvature in two distinct ways. One extension asymptotically connects with centro-affine geometry on convex bodies as curvature approaches zero, while the other converges with Euclidean geometry. Notably, our newly introduced curvature entropy for spherical convex bodies emerges as a natural counterpart to both the centro-affine entropy and the Gaussian entropy of convex bodies in Euclidean space., Comment: 43 pages

The purpose of this paper is to introduce the new concept of weighted floating functions associated with log concave or $s$-concave functions. This leads to new notions of weighted functional affine surface areas. Their relation to more traditional versions of functional affine surface areas as well as to the classical affine surface areas for convex bodies is discussed in detail., Comment: arXiv admin note: text overlap with arXiv:1711.11088

We investigate a natural analog to Lutwak's $p$-affine surface area in $d$-dimensional spherical, hyperbolic and de Sitter space. In particular, we show that these curvature measures appear naturally as the volume derivative of floating bodies of non-Euclidean convex bodies conjugated by duality, such as spherical, hyperbolic and de Sitter convex bodies. We provide a unifying framework by establishing a real-analytic version of this relation controlled by the constant curvature of the $d$-dimensional real space form. These new curvature measures relate in two distinctly different ways to curvature measures on Euclidean space, one of which is Lutwak's centro-affine invariant $p$-affine surface area, and the other is related to a rigid-motion invariant curvature measure that appears naturally as the volume derivative of Schneider's mean-width separation body., Comment: 40 pages, 9 figures

We investigate the weighted $L_p$ affine surface areas which appear in the recently established $L_p$ Steiner formula of the $L_p$ Brunn Minkowski theory. We show that they are valuations on the set of convex bodies and prove isoperimetric inequalities for them. We show that they are related to $f$ divergences of the cone measures of the convex body and its polar, namely the Kullback-Leibler divergence and the R\'enyi-divergence.

We give an overview of the affine surface area, its properties and its history.

Inspired by an $L_p$ Steiner formula for the $L_p$ affine surface area proved by Tatarko and Werner, we define, in analogy to the classical Steiner formula, $L_p$-Steiner quermassintegrals. Special cases include the classical mixed volumes, the dual mixed volumes, the $L_p$ affine surface areas and the mixed $L_p$ affine surface areas. We investigate the properties of the $L_p$-Steiner quermassintegrals in a special class of convex bodies. In particular, we show that they are rotation and reflection invariant valuations in this class of convex bodies with a certain degree of homogeneity. Such valuations seem new and have not been observed before.

In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.

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 Brunn Minkowski theory are special cases of this new Steiner formula. This new Steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature. They have the intrinsic volumes of the classical Brunn Minkowski theory and the dual quermassintegrals of the dual Brunn Minkowski theory as well as special cases. Properties of these new quantities are investigated, a connection to information theory among them. A Steiner formula for the $s$-th mixed $L_p$ affine surface area of a Minkowski outer parallel body for any real parameters $p$ and~$s$ is also given.

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This leads to new asymptotic results for polytopes in these spaces. We also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in Euclidean space.

Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floating areas are combined to derive new asymptotic approximation results on the sphere, in hyperbolic space and in Hilbert geometries.

We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space. Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies.