Completeness in affine and statistical geometry

We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.Comment: 15 page

Bochner's technique for statistical structures

The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge's theory, Bochner-Weitzenbock and Simon's type formulas. Moreover, a few global and local theorems on the geometry of statistical structures are proved, for instance, theorems saying that under some topological and geometrical conditions a statistical structure must be trivial. We also introduce a new concept of sectional curvature depending on statistical connections. On the base of this notion we study the curvature operator and prove some analogues of well-known theorems from Riemannian geometry

Curvature bounded conjugate symmetric statistical structures with complete metric

In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the curvature satisfies some inequalities

Completeness in affine and statistical geometry

We begin the study of completeness of affine connections, especially those on statistical manifolds or on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties

Curvature bounded conjugate symmetric statistical structures with complete metric

In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the curvature satisfies some inequalities.Comment: 15 page

α\alpha-connections in generalized geometry

We consider a family of $\alpha$-connections defined by a pair of generalized dual quasi-statistical connections $(\hat{\nabla},\hat{\nabla}^*)$ on the generalized tangent bundle $(TM\oplus T^*M, \check{h})$ and determine their curvature, Ricci curvature and scalar curvature. Moreover, we provide the necessary and sufficient condition for $\hat \nabla^*$ to be an equiaffine connection and we prove that if $h$ is symmetric and $\nabla h=0$, then $(TM\oplus T^*M, \check{h}, \hat{\nabla}^{(\alpha)}, \hat{\nabla}^{(-\alpha)})$ is a conjugate Ricci-symmetric manifold. Also, we characterize the integrability of a generalized almost product, of a generalized almost complex and of a generalized metallic structure w.r.t. the bracket defined by the $\alpha$-connection. Finally we study $\alpha$-connections defined by the twin metric of a pseudo-Riemannian manifold, $(M,g)$, with a non-degenerate $g$-symmetric $(1,1)$-tensor field $J$ such that $d^\nabla J=0$, where $\nabla$ is the Levi-Civita connection of $g$.Comment: 29 page

Centro-Affine Differential Geometry and the Log-Minkowski Problem

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 centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in $\mathbb{R}^n$ is a centro-affine unit-sphere, it has constant centro-affine Ricci curvature equal to $n-2$, in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn-Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the $L^p$- and log-Minkowski problems, as well as the corresponding global $L^p$- and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body $\bar K$ in $\mathbb{R}^n$, there exists an origin-symmetric convex body $K$ with $\bar K \subset K \subset 8 \bar K$, so that $K$ satisfies the log-Minkowski conjectured inequality, and so that $K$ is uniquely determined by its cone-volume measure $V_K$. If $\bar K$ is not extremely far from a Euclidean ball to begin with, analogous isometric results are derived as well.Comment: 63 pages; rewrote introduction, adding the preliminary Theorem 1.1 and the more expanded Theorem 2.1 regarding the equivalence of the various variants of the conjectur

A Bayesian construction of asymptotically unbiased estimators

A differential geometric framework to construct an asymptotically unbiased estimator of a function of a parameter is presented. The derived estimator asymptotically coincides with the uniformly minimum variance unbiased estimator, if a complete sufficient statistic exists. The framework is based on the maximum a posteriori estimation, where the prior is chosen such that the estimator is unbiased. The framework is demonstrated for the second-order asymptotic unbiasedness (unbiased up to $O(n^{-1})$ for a sample of size $n$). The condition of the asymptotic unbiasedness leads the choice of the prior such that the departure from a kind of harmonicity of the estimand is canceled out at each point of the model manifold. For a given estimand, the prior is given as an integral. On the other hand, for a given prior, we can address the bias of what estimator can be reduced by solving an elliptic partial differential equation. A family of invariant priors, which generalizes the Jeffreys prior, is mentioned as a specific example. Some illustrative examples of applications of the proposed framework are provided.Comment: 28 pages, 2 figure