An induction proof of the generalized Blaschke-Petkantschin formula

1996 ◽  
Vol 28 (2) ◽  
pp. 334-334
Author(s):  
E. B. Vedel Jensen
Keyword(s):  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Integral Geometry ◽  
Geometric Measure ◽  
Measure Decomposition

The classical Blaschke-Petkanschin formula is a formula in integral geometry givmg a geometric measure decomposition of the q-fold product of Lebesgue measure. The original versions are due to Blaschke and Petkanschin in the 1930s. In Zähle (1990) and Jensen and Kiêu (1992), generalized versions have been derived, where Lebesgue measure is replaced by Hausdorff measure.

An induction proof of the generalized Blaschke-Petkantschin formula

1996 ◽  
Vol 28 (02) ◽  
pp. 334
Author(s):  
E. B. Vedel Jensen
Keyword(s):  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Integral Geometry ◽  
Geometric Measure ◽  
Measure Decomposition

The classical Blaschke-Petkanschin formula is a formula in integral geometry givmg a geometric measure decomposition of the q-fold product of Lebesgue measure. The original versions are due to Blaschke and Petkanschin in the 1930s. In Zähle (1990) and Jensen and Kiêu (1992), generalized versions have been derived, where Lebesgue measure is replaced by Hausdorff measure.

scholarly journals POLAR DECOMPOSITION OF THE k-FOLD PRODUCT OF LEBESGUE MEASURE ON ℝn

2012 ◽  
Vol 85 (2) ◽  
pp. 315-324 ◽  
Author(s):  
S. REZA MOGHADASI
Keyword(s):  
Lebesgue Measure ◽  
Polar Decomposition ◽  
Rotational Symmetry ◽  
Orthogonal Transformation ◽  
Geometric Measure ◽  
Measure Decomposition

AbstractThe Blaschke–Petkantschin formula is a geometric measure decomposition of the q-fold product of Lebesgue measure on ℝn. Here we discuss another decomposition called polar decomposition by considering ℝn×⋯×ℝn as ℳn×k and using its polar decomposition. This is a generalisation of the Blaschke–Petkantschin formula and may be useful when one needs to integrate a function g:ℝn×⋯×ℝn→ℝ with rotational symmetry, that is, for each orthogonal transformation O,g(O(x1),…,O(xk))=g(x1,…xk). As an application we compute the moments of a Gaussian determinant.

scholarly journals A mass transference principle for systems of linear forms and its applications

2018 ◽  
Vol 154 (5) ◽  
pp. 1014-1047 ◽  
Author(s):  
Demi Allen ◽  
Victor Beresnevich
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
Linear Forms ◽  
Transference Principle ◽  
Integer Points ◽  
Additional Constraints

In this paper we establish a general form of the mass transference principle for systems of linear forms conjectured in 2009. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine–Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine–Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira.

On the connexion between Hausdorff measures and generalized capacity

1961 ◽  
Vol 57 (3) ◽  
pp. 524-531 ◽  
Author(s):  
S. J. Taylor
Keyword(s):  
Euclidean Space ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Real Function ◽  
Function Theory ◽  
Logarithmic Capacity ◽  
Hausdorff Measures ◽  
Positive Capacity ◽  
Special Case

For any real function h(t) which is continuous and monotonic increasing for t > 0 with , Hausdorff (10) in 1918 denned a Carathéodory measure with respect to h(t) which has subsequently been known as Hausdorff measure. For analysing sets in Euclidean space, these measures have proved both useful and interesting. Given a real function Φ(t) which is continuous and monotonic decreasing for t > 0 with , Frostman(9) in 1935 denned capacity with respect to Φ(t). Lebesgue measure in Euclidean k-space is a special case of Hausdorff measure, and capacity with respect to Φ(t) becomes logarithmic capacity or Newtonian capacity in the cases , Φ(t)=1/t, respectively. The interrelationship between h-measure and Φ-capacity has been of interest in both directions: (i) in applications to function theory one may be able to determine whether or not a set has positive capacity by examining the h-measure for suitable h(t) (see, for example, (5)); (ii) it may be possible to determine the measure properties of a set from knowledge of its capacity (see, for example, (7) and (17)).

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

scholarly journals On Perimeters and Volumes of Fattened Sets

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Andrea C. G. Mennucci
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Coarea Formula ◽  
Compact Set ◽  
Geometric Measure ◽  
General Bound

In this paper we analyze the shape of fattened sets; given a compact set C⊂RN let Cr be its r-fattened set; we prove a general bound rP(Cr)≤NL({Cr∖C}) between the perimeter of Cr and the Lebesgue measure of Cr∖C. We provide two proofs: one elementary and one based on Geometric Measure Theory. Note that, by the Flemin–Rishel coarea formula, P(Cr) is integrable for r∈(0,a). We further show that for any integrable continuous decreasing function ψ:(0,1/2)→(0,∞) there exists a compact set C⊂RN such that P(Cr)≥ψ(r). These results solve a conjecture left open in (Mennucci and Duci, 2015) and provide new insight in applications where the fattened set plays an important role.

scholarly journals A note on pointwise convergence for the Schrödinger equation

2017 ◽  
Vol 166 (2) ◽  
pp. 209-218 ◽  
Author(s):  
RENATO LUCÀ ◽  
KEITH M. ROGERS
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Schrodinger Equation ◽  
Pointwise Convergence

AbstractWe consider Carleson's problem regarding pointwise convergence for the Schrödinger equation. Bourgain proved that there is initial data, in Hs(ℝn) with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set of nonzero Lebesgue measure. We provide a different example enabling the generalisation to fractional Hausdorff measure.

scholarly journals A remark on boundedness of Bloch functions

1991 ◽  
Vol 44 (3) ◽  
pp. 527-528 ◽  
Author(s):  
Krzysztof Samotij
Keyword(s):  
Holomorphic Function ◽  
Unit Circle ◽  
Lebesgue Measure ◽  
Unit Disk ◽  
Complex Plane ◽  
Hausdorff Measure ◽  
Bloch Space ◽  
Bloch Functions ◽  
Zero Measure

Two consequences of a theorem of Dahlberg are derived. Let f be a holomorphic function in the unit disk D of the complex plane, and let E be an Fσ subset of the unit circle T. Suppose that |f(rw)| ≤ M, ω ∈ T/E, for some constant M.Then f is bounded in either of the two cases:(i) if f is in the Bloch space and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log log (2πee/t),(ii) if f is integrable with respect to the planar Lebesgue measure on D and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log(2πee/t).

scholarly journals Isoperimetric and Functional Inequalities

2018 ◽  
Vol 25 (3) ◽  
pp. 331-342
Author(s):  
Vladimir S. Klimov
Keyword(s):  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Integral Functional ◽  
Minkowski Inequality ◽  
Variational Problems ◽  
Functional Inequalities ◽  
Dimensional Hausdorff Measure ◽  
The Third ◽  
Differentiable Functions ◽  
Dimensional Lebesgue Measure

We establish lower estimates for an integral functional$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$where \(\Omega\) -- a bounded domain in \(\mathbb{R}^n \; (n \geqslant 2)\), an integrand \(f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)\) -- a function that is \(B\)-measurable with respect to a variable \(t\) and is convex and even in the variable \(p\), \(\nabla u(x)\) -- a gradient (in the sense of Sobolev) of the function \(u \colon \Omega \rightarrow \mathbb{R}\). In the first and the second sections we utilize properties of permutations of differentiable functions and an isoperimetric inequality \(H^{n-1}( \partial A) \geqslant \lambda(m_n A)\), that connects \((n-1)\)-dimensional Hausdorff measure \(H^{n-1}(\partial A )\) of relative boundary \(\partial A\) of the set \(A \subset \Omega\) with its \(n\)-dimensional Lebesgue measure \(m_n A\). The integrand \(f\) is assumed to be isotropic, i.e. \(f(t,p) = f(t,q)\) if \(|p| = |q|\).Applications of the established results to multidimensional variational problems are outlined. For functions \( u \) that vanish on the boundary of the domain \(\Omega\), the assumption of the isotropy of the integrand \( f \) can be omitted. In this case, an important role is played by the Steiner and Schwartz symmetrization operations of the integrand \( f \) and of the function \( u \). The corresponding variants of the lower estimates are discussed in the third section. What is fundamentally new here is that the symmetrization operation is applied not only to the function \(u\), but also to the integrand \(f\). The geometric basis of the results of the third section is the Brunn-Minkowski inequality, as well as the symmetrization properties of the algebraic sum of sets.

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