Affine invariant points and maps for sets were introduced by Grunbaum to study the symmetry structure of convex sets. We extend these notions to a functional setting. The role of symmetry of the set is now taken by evenness of the function. We show t
hat among the examples for affine invariant points are the classical center of gravity of a log-concave function and its Santalo point. We also show that the recently introduced floating functions and the John- and Lowner functions are examples of affine invariant maps. Their centers provide new examples of affine invariant points for log-concave functions.
Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke-Santalo inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an ent
ropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a pointwise Prekopa-Leindler inequality and show a monotonicity property of the multimarginal Blaschke-Santalo functional.
We introduce the notion of Loewner (ellipsoid) function for a log concave function and show that it is an extension of the Loewner ellipsoid for convex bodies. We investigate its duality relation to the recently defined John (ellipsoid) function by A
lonso-Gutierrez, Merino, Jimenez and Villa. For convex bodies, John and Loewner ellipsoids are dual to each other. Interestingly, this need not be the case for the John function and the Loewner function.
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 Lutwaks dual Brunn Minkowski
theory are special cases of this new Steiner formula. This new Steiner formula and its localiz
The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of
the involved sets, more precisely, by what is called the surface area deviation. The proof uses arguments and constructions from probability, convex and integral geometry. The bound is closely related to $p$-affine surface areas.
Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stan
ds as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.
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 stru
cture. 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.
We study a new construction of bodies from a given convex body in $mathbb{R}^{n}$ which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to $p$-affine surface areas. We sho
w that these bodies are related to Ulams long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.
We introduce floating bodies for convex, not necessarily bounded subsets of $mathbb{R}^n$. This allows us to define floating functions for convex and log concave functions and log concave measures. We establish the asymptotic behavior of the integral
difference of a log concave function and its floating function. This gives rise to a new affine invariant which bears striking similarities to the Euclidean affine surface area.
Let $K$ be a convex body in $mathbb{R}^n$ and $f : partial K rightarrow mathbb{R}_+$ a continuous, strictly positive function with $intlimits_{partial K} f(x) d mu_{partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmet
ric difference metric by an arbitrarily positioned polytope $P_f$ in $mathbb{R}^n$ having a fixed number of vertices. This generalizes a result by Ludwig, Schutt and Werner $[36]$. The polytope $P_f$ is obtained by a random construction via a probability measure with density $f$. In our result, the dependence on the number of vertices is optimal. With the optimal density $f$, the dependence on $K$ in our result is also optimal.
