The (π, π)-th dual curvature measure was introduced by Lutwak, Yang and Zhang. In this paper, we study the πΏ π cosine transform of the (π, π)-th dual curvature measure which defines a new convex body. We prove that the new convex body unifies the πΏ π Petty projection body and the πΏ π centroid body. The corresponding affine isoperimetric inequality is also obtained. It is an extension of the known πΏ π Petty projection inequality.
K E Y W O R D SπΏ π centroid body, πΏ π cosine transform, πΏ π Petty projection body, (π, π)-th dual curvature measure M S C ( 2 0 2 0 ) 52A20, 52A40
The Wulff isoperimetric inequality is a natural extension of the classical isoperimetric inequality (Green and Osher in Asian J. Math. 3:659-676 1999). In this paper, we establish some Bonnesen-style Wulff isoperimetric inequalities and reverse Bonnesen-style Wulff isoperimetric inequalities. Those inequalities obtained are extensions of known Bonnesen-style inequalities and reverse Bonnesen-style inequalities.
Let Ξ³ be a closed strictly convex curve in the Euclidean plane R 2 with length L and enclosing an area A, andΓ 1 denote the oriented area of the domain enclosed by the locus of curvature centers of Ξ³ . Pan and Xu conjectured that there exists a best constant C such thatwith equality if and only if Ξ³ is a circle. In this paper, we give an affirmative answer to this question. Moreover, instead of working with the domain enclosed by the locus of curvature centers we consider the domain enclosed by the locus of width centers of Ξ³ , and we obtain some new reverse isoperimetric inequalities.
MSC: 52A10; 52A22
Inspired by the equivalence between isoperimetric inequality and Sobolev inequality, we provide a new connection between geometry and analysis. We define the minimal perimeter of a log-concave function and establish a characteristic theorem of this extremal problem for log-concave functions analogous to convex bodies.
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