Measuring the Surface Area of a Convex Body

1944 ◽  
Vol 45 (4) ◽  
pp. 793 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Convex Body ◽  
Surface Area

scholarly journals Inequalities for the Surface Area of Projections of Convex Bodies

2018 ◽  
Vol 70 (4) ◽  
pp. 804-823 ◽  
Author(s):  
Apostolos Giannopoulos ◽  
Alexander Koldobsky ◽  
Petros Valettas
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
Maximal Surface ◽  
Projection Body ◽  
Projections Of Convex Bodies ◽  
Lower Dimensional

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

scholarly journals ISOPERIMETRIC INEQUALITIES FOR Lp GEOMINIMAL SURFACE AREA

2011 ◽  
Vol 53 (3) ◽  
pp. 717-726 ◽  
Author(s):  
BAOCHENG ZHU ◽  
NI LI ◽  
JIAZU ZHOU
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Type Inequality ◽  
Isoperimetric Inequalities ◽  
Surface Areas ◽  
Cyclic Inequality ◽  
Geominimal Surface Area

AbstractIn this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke–Santaló type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.

Asymptotic Formulae for the Lattice Point Enumerator

1999 ◽  
Vol 51 (2) ◽  
pp. 225-249 ◽  
Author(s):  
U. Betke ◽  
K. Böröczky
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Positive Curvature ◽  
Convex Bodies ◽  
Mixed Volume ◽  
Lattice Points ◽  
Asymptotic Formulae ◽  
General Convex ◽  
Lattice Surface ◽  
Small Deformations

AbstractLet M be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large λ the number of lattice points in λM is given by G(λM) = V(λM) + O(λd−1−ε(d)) for some positive ε(d). Here we give for general convex bodies the weaker estimatewhere SZd (M) denotes the lattice surface area of M. The term SZd is optimal for all convex bodies and o(λd−1) cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of M.Further we deal with families {Pλ} of convex bodies where the only condition is that the inradius tends to infinity. Here we havewhere the convex body K satisfies some simple condition, V(Pλ; K; 1) is some mixed volume and S(Pλ) is the surface area of Pλ.

scholarly journals Characterizations of central symmetry for convex bodies in Minkowski spaces

2009 ◽  
Vol 46 (4) ◽  
pp. 493-514
Author(s):  
Gennadiy Averkov ◽  
Endre Makai ◽  
Horst Martini
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Analogous Result ◽  
Area Measure ◽  
Minkowski Metric ◽  
Surface Area Measure ◽  
Shadow Boundary ◽  
Dimensional Minkowski Space ◽  
Centrally Symmetric ◽  
Definition Of

K. Zindler [47] and P. C. Hammer and T. J. Smith [19] showed the following: Let K be a convex body in the Euclidean plane such that any two boundary points p and q of K , that divide the circumference of K into two arcs of equal length, are antipodal. Then K is centrally symmetric. [19] announced the analogous result for any Minkowski plane \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^2$$ \end{document}, with arc length measured in the respective Minkowski metric. This was recently proved by Y. D. Chai — Y. I. Kim [7] and G. Averkov [4]. On the other hand, for Euclidean d -space ℝ d , R. Schneider [38] proved that if K ⊂ ℝ d is a convex body, such that each shadow boundary of K with respect to parallel illumination halves the Euclidean surface area of K (for the definition of “halving” see in the paper), then K is centrally symmetric. (This implies the result from [19] for ℝ 2 .) We give a common generalization of the results of Schneider [38] and Averkov [4]. Namely, let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^d$$ \end{document} be a d -dimensional Minkowski space, and K ⊂ \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^d$$ \end{document} be a convex body. If some Minkowskian surface area (e.g., Busemann’s or Holmes-Thompson’s) of K is halved by each shadow boundary of K with respect to parallel illumination, then K is centrally symmetric. Actually, we use little from the definition of Minkowskian surface area(s). We may measure “surface area” via any even Borel function ϕ: Sd −1 → ℝ, for a convex body K with Euclidean surface area measure dSK ( u ), with ϕ( u ) being dSK ( u )-almost everywhere non-0, by the formula B ↦ ∫ B ϕ( u ) dSK ( u ) (supposing that ϕ is integrable with respect to dSK ( u )), for B ⊂ Sd −1 a Borel set, rather than the Euclidean surface area measure B ↦ ∫ BdSK ( u ). The conclusion remains the same, even if we suppose surface area halving only for parallel illumination from almost all directions. Moreover, replacing the surface are a measure dSK ( u ) by the k -th area measure of K ( k with 1 ≦ k ≦ d − 2 an integer), the analogous result holds. We follow rather closely the proof for ℝ d , which is due to Schneider [38].

scholarly journals Estimating volume and surface area of a convex body via its projections or sections

2019 ◽  
Vol 244 (3) ◽  
pp. 245-264 ◽  
Author(s):  
Alexander Koldobsky ◽  
Christos Saroglou ◽  
Artem Zvavitch
Keyword(s):  
Convex Body ◽  
Surface Area

scholarly journals Lp−Winterniz problem on firey projection of convex bodies

2014 ◽  
Vol 45 (2) ◽  
pp. 179-193
Author(s):  
Tong Yi MA ◽  
Li Li Zhang
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
Affine Surface ◽  
Projection Body ◽  
Affine Surface Area ◽  
Funk Transform

For $p\geq 1$, Lutwak, Yang and Zhang introduced the concept of $p$-projection body, and Lutwak introduced the concept of $L_{p}-$ affine surface area of convex body. In this paper, we develop the Minkowski-Funk transform approach in the $L_{p}$-Brunn-Minkowski theory. We consider the question of whether $\Pi_{p}K\subseteq \Pi_{p}L$ implies $\Omega_{p}(K) \leq \Omega_{p}(L)$, where $\Pi_{p}K$ and $\Omega_{p}K$ denotes the $p-$projection body of convex body $K$ and the $L_{p}-$affine surface area of convex body $K$, respectively. We also formulate and solve a generalized $L_{p}-$Winterniz problem for Firey projections.

scholarly journals Geometric Modeling of Medical Filtered Back-Projection Images

2020 ◽  
Vol 8 (5) ◽  
Author(s):  
Atefeh Hasan-Zadeh
Keyword(s):  
Medical Imaging ◽  
Convex Body ◽  
Surface Area ◽  
Geometric Modeling ◽  
Symplectic Geometry ◽  
Finsler Geometry ◽  
Back Projection ◽  
Filtered Back Projection ◽  
Time Frequency ◽  
Projection Image

This paper deals with the geometric concepts of medical imaging. The main result is proving this novel achievement that the time-frequency projections from some angles in the filtered back-projection image can be calculated by the surface area of the convex body and symplectic Holmes-Thompson volume. For this purpose, the necessary notions are introduced by the surface area of a convex body of the corresponding projective spaces, Radon and Fourier transform in integral geometry, Holmes-Thompson measure in Finsler-Minkowsi-symplectic geometry. This result leads to the other correspondence between basic notions of medical imaging such as the projections from some angles which are equally spaced, the image-frequency of these projections, the low image correctness, and the corresponding notions of Finsler geometry.

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