Related papers: The Wulff construction for convex integrands

For given continuous functions $\gamma_{{}_{i}}: S^{n}\to \mathbb{R}_{+}$ (where $i=1, 2$), the functions $\gamma_{{}_{max}}$ and $\gamma_{{}_{min}}$ can be defined as natural way. In this paper, we show that the Wulff shape associated to…

Let $\gamma: S^n\to \mathbb{R}_+$ be a convex integrand and $\mathcal{W}_\gamma$ be the Wulff shape of $ \gamma$. Apex point naturally arise in non-smooth Wulff shape, in particular, vertex of convex polytope. %Let $P\in S^n$. In this…

Let $\gamma: S^n\to \mathbb{R}_+$ be a continuous function and let $\mathcal{W}_\gamma$ be the Wulff shape associated with $\gamma$. In this paper, we show that if the boundary of $\mathcal{W}_\gamma$ is a $C^1$ submanifold, then $\gamma$…

In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class $C^1$. Moreover, applications of this result are given.

For a function $f$ which foliates a one-sided neighbourhood of a closed hypersurface $M$, we give an estimate of the distance of $M$ to a Wulff shape in terms of the $L^{p}$-norm of the traceless $F$-Hessian of $f$, where $F$ is the support…

For any Wulff shape, its dual Wulff shape is naturally defined. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, it is shown that a Wulff shape is self-dual if and only if the spherical convex…

The Wills functional $\mathcal{W}(K)$ of a convex body $K$, defined as the sum of its intrinsic volumes $\mathrm{V}_i(K)$, turns out to have many interesting applications and properties. In this paper we make profit of the fact that it can…

For any Wulff shape $\mathcal{W}$, its dual Wulff shape and spherical Wulff shape $\widetilde{\mathcal{W}}$ can be defined naturally. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, we show…

Given a strictly increasing, continuous function $\vartheta:\R_+\to\R_+$, based on the cost functional $\int_{X\times X}\vartheta(d(x,y))\,d q(x,y)$, we define the $L^\vartheta$-Wasserstein distance $W_\vartheta(\mu,\nu)$ between…

The present work revisits the classical Wulff problem restricted to crystalline integrands, a class of surface energies that gives rise to finitely faceted crystals. The general proof of the Wulff theorem was given by J.E. Taylor (1978) by…

Let $H(D)$ denote the space of holomorphic functions on the unit disk $D$. We characterize those radial weights $w$ on $D$, for which there exist functions $f, g \in H(D)$ such that the sum $|f| + |g|$ is equivalent to $w$. Also, we obtain…

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the…

The width $w$ of a curve $\gamma$ in Euclidean space $R^n$ is the infimum of the distances between all pairs of parallel hyperplanes which bound $\gamma$, while its inradius $r$ is the supremum of the radii of all spheres which are…

The classical ``$H=W$" theorem establishes the identity between two function spaces on an arbitrary nonempty open set in the Euclidean spaces: the space $W$ defined via weak derivatives, and the space $H$ defined as the closure of smooth…

We make some remarks on the global shape of continuous convex functions defined on a Banach space $Z$. Among other results we prove that if $Z$ is separable then for every continuous convex function $f:Z\to\mathbb{R}$ there exist a unique…

We view all smooth metrics $g$ on a closed surface $\Sigma$ through their Nash isometric embeddings $f_g: (\Sigma,g) \rightarrow (\mathbb{S}^{\tilde{n}}, \tilde{g})$ into a standard sphere of large, but fixed, dimension $\tilde{n}$. We…

Given a positive function F on S n satisfying an appropriate con-vexity assumption, we consider hypersurfaces for which a linear combination of some higher order anisotropic curvatures is constant. We define the varia-tional problem for…

We study necessary and sufficient conditions for a Muckenhoupt weight $w \in L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $u \in…

It is shown that if $\gamma$ is a path of finite $p$ variation ($1\leq p< 2$) in a euclidean vector space and $f,g,h$ are Lipschitz functions on the trace of $\gamma$ then $s\mapsto F(s)=\int_\gamma f^sg dh$ defines an entire holomorphic…

Let $\mathcal{H}ol(B_d)$ denote the space of holomorphic functions on the unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. Given a log-convex strictly positive weight $w(r)$ on $[0,1)$, we construct a function $f\in\mathcal{H}ol(B_d)$ such that…