We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature ${ℳ}_p^{\alpha }\left(X\right):={\int }_X{\int }_X{\int }_X{\kappa }^p(x,y,z)d_X^{\alpha }\left(x\right)d_X^{\alpha }\left(y\right)d_X^{\alpha }\left(z\right)$, where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that ${ℳ}_p^{\alpha }\left(X\right)<\infty$ for p ≥ 3α implies the existence of a weak approximate α-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant case p = 3 for...

A characterization of Haar null sets in the sense of Christensen is given. Using it, we show that if the dual of a Banach space $X$ has the Banach-Saks property, then closed and convex subsets of $X$ with empty interior are Haar null.

We construct a differentiable function $f:{\mathbf{R}}^n\to \mathbf{R}$ ($n\ge 2$) such that the set ${\left(\nabla f\right)}^{-1}\left(B(0,1)\right)$ is a nonempty set of Hausdorff dimension $1$. This answers a question posed by Z. Buczolich.

This is an expository paper dealing with Jan Marik's results concerning perimeter and the divergence theorem of Gauss-Green-Ostrogradski.

We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety $M\subset {ℝ}^n$ is dense in the Hilbert space $L^2(M,e^{-{\left|x\right|}^2}d\mu )$, where dμ denotes the volume form of M and $d\nu =e^{-{\left|x\right|}^2}d\mu$ the Gaussian measure on M.

We study the Hausdorff lower semicontinuous envelope of the length in the plane. This envelope is taken with respect to the Hausdorff metric on the space of the continua. The resulting quantity appeared naturally as the rate function of a large deviation principle in a statistical mechanics context and seems to deserve further analysis. We provide basic simple results which parallel those available for the perimeter of Caccioppoli and De Giorgi.

We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the...

Doubling measures appear in relation to quasiconformal mappings of the unit disk of the complex plane onto itself. Each such map determines a homeomorphism of the unit circle on itself, and the problem arises, which mappings f can occur as boundary mappings?

We show that any strictly mean convex translator of dimension n ≥ 3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n ≥ 3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular...