Related papers: When is a Minkowski norm strictly sub-convex?

If a pair of subsets of two-dimensional Euclidean space nearly achieves equality in the Brunn-Minkowski inequality, in the sense that the measure of the associated sumset is nearly equal to the lower bound provided by the inequality, then…

We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above…

We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that, given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In…

It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let $X$ be a locally convex Hausdorff space and $f \colon X \mapsto \left( - \infty , \infty \right]$ a function with compact…

It is shown that for any sufficiently regular even Minkowski valuation $\Phi$ which is homogeneous and intertwines rigid motions, and for any convex body $K$ in a smooth neighborhood of the unit ball, there exists a sequence of positive…

The uniqueness of the $L_p$-Minkowski problem has been a long standing problem in convex geometry. In the groundbreaking paper by Brendle-Choi-Daskalopoulos (Acta Math, {\bf219}, 2017), a full uniqueness result was shown for the subcritical…

We investigate Minkowski additive, continuous, and translation invariant operators $\Phi:\mathcal{K}^n\to\mathcal{K}^n$ defined on the family of convex bodies such that the volume of the image $\Phi(K)$ is bounded from above and below by…

In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disk. We also estimate the coefficient bound and obtain growth, covering and area theorems for…

In light of the log-Brunn-Minkowski conjecture, various attempts have been made to define the geometric mean of convex bodies. Many of these constructions are fairly complex and/or fail to satisfy some natural properties one would expect of…

On a convex bounded open set, we prove that Poincar\'e-Sobolev constants for functions vanishing at the boundary can be bounded from below in terms of the norm of the distance function in a suitable Lebesgue space. This generalizes a result…

Several open problems concerning Minkowski endomorphisms and Minkowski valuations are solved. More precisely, it is proved that all Minkowski endomorphisms are uniformly continuous but that there exist Minkowski endomorphisms that are not…

For the minimal graph with strict convex level sets, we find an auxiliary function to study the Gaussian curvature of the level sets. We prove that this curvature function is a concave function with respect to the height of the minimal…

In Guo and Peng's article [Spherically convex sets and spherically convex functions, J. Convex Anal. 28 (2021), 103--122], one defines the notions of spherical convex sets and functions on "general curved surfaces" in $\mathbb{R}^{n}$…

In this paper we explore questions regarding the Minkowski sum of the boundaries of convex sets. Motivated by a question suggested to us by V.~Milman regarding the volume of $\partial K+ \partial T$ where $K$ and $T$ are convex bodies, we…

The Minkowski Question Mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only…

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike…

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a…

Let $m\in\mathbb{N},$ $m\geq 2,$ and let $\{p_j\}_{j=1}^m$ be a finite subset of $\mathbb{S}^2$ such that $0\in\mathbb{R}^3$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the…

In this paper, inspired by the work of Spruck-Xiao [27] and based partly on a result of Derdzi\'nski [11], we prove the convexity of complete 2-convex translating and expanding solitons to the mean curvature flow in $\mathbb{R}^{n+1}$. More…

Let us define, for a compact set $A \subset \mathbb{R}^n$, the Minkowski averages of $A$: $$ A(k) = \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots +…