Abstract The Brunn-Minkowski theory in convex geometry concerns, among other things, the
volumes, mixed volumes, and surface area measures of convex bodies. We study …

The Brunn-Minkowski Theory has seen several generalizations over the past century. Many
of the core ideas have been generalized to measures. With the goal of framing these …

Sharp affine fractional Sobolev inequalities for functions on $\mathbb R^ n $ are
established. For each $0< s< 1$, the new inequalities are significantly stronger than (and …

In 1970, Schneider generalized the difference body of a convex body to higher-order, and
also established the higher-order analogue of the Rogers-Shephard inequality. In this …

The inequality of Berwald is a reverse-H\" older like inequality for the $ p $ th average, $ p\in
(-1,\infty), $ of a non-negative, concave function over a convex body in $\mathbb {R}^ n …

In 2019, Livshyts studied the Minkowski problem of measures in ℝ n with positive
homogeneous and positive concave density functions. After that, Wu studied the L p …

The L p John ellipsoids for general measures are proposed, which aims to extend the L p
John ellipsoids introduced by Lutwak, Yang, Zhang in their most possible general settings …

For a convex body $ K $ in $\mathbb {R}^ n $, the inequalities of Rogers-Shephard and
Zhang, written succinctly, are $\text {vol} _n (DK)\leq\binom {2n}{n}\text {vol} _n (K)\leq\text …

We characterize rotation equivariant bounded linear operators from $ C (\mathbb {S}^{n-1})
$ to $ C^ 2 (\mathbb {S}^{n-1}) $ by the mass distribution of the spherical Laplacian of their …

The inequality of Berwald is a reverse-Hölder like inequality for the pth average, p∈(− 1,∞),
of a non-negative, concave function over a convex body in Rn. We prove Berwald's …