A surface area formula for compact hypersurfaces in Rn

The classical Cauchy surface area formula states that the surface area of the boundary ∂ K = Σ of any n -dimensional convex body in the n -dimensional Euclidean space R n can be obtained by the average of the projected areas of Σ along all directions in S n − 1 . In this note, we generalize the form...

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Bibliographic Details
Published inJournal of inequalities and applications Vol. 2024; no. 1; p. 44
Main Author Huang, Yen-Chang
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2024
Springer Nature B.V
Subjects
Analysis
Applications of Mathematics
Euclidean geometry
Euclidean space
Geometry
Hyperspaces
Lie groups
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
Neighborhoods
Subspaces
Surface area
53C17
Projection
Cauchy surface area formula
Euclidean spaces
53C23
53C65
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Summary:The classical Cauchy surface area formula states that the surface area of the boundary ∂ K = Σ of any n -dimensional convex body in the n -dimensional Euclidean space R n can be obtained by the average of the projected areas of Σ along all directions in S n − 1 . In this note, we generalize the formula to the boundary of arbitrary n -dimensional submanifold in R n by introducing a natural notion of projected areas along any direction in S n − 1 . This surface area formula derived from the new notion coincides with not only the result of the Crofton formula but also with that of De Jong (Math. Semesterber. 60(1):81–83, 2013 ) by using a tubular neighborhood. We also define the projected r -volumes of Σ onto any r -dimensional subspaces and obtain a recursive formula for mean projected r -volumes of Σ.
Bibliography:ObjectType-Article-1
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ISSN:1025-5834
1029-242X
DOI:10.1186/s13660-024-03129-x