scholarly journals General Blaschke Bodies and the Asymmetric Negative Solutions of Shephard Problem

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 610
Author(s):  
Tian Li ◽  
Weidong Wang ◽  
Yaping Mao
Keyword(s):  
Surface Area ◽  
Convex Bodies ◽  
Affine Surface ◽  
Affine Surface Area ◽  
Extremal Values ◽  
Shephard Problem ◽  
Projection Bodies

In this article, based on the Blaschke combination of convex bodies, we define the general Blaschke bodies and obtain the extremal values of their volume and affine surface area. Further, we study the asymmetric negative solutions of the Shephard problem for the projection bodies.

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 A floating body approach to Fefferman's hypersurface measure

2006 ◽  
Vol 98 (1) ◽  
pp. 69 ◽  
Author(s):  
David E. Barrett
Keyword(s):  
Surface Area ◽  
Floating Body ◽  
Affine Surface ◽  
Affine Surface Area

The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.

scholarly journals Functional Versions ofLp-Affine Surface Area and Entropy Inequalities

2015 ◽  
Vol 2016 (4) ◽  
pp. 1223-1250 ◽  
Author(s):  
Umut Caglar ◽  
Matthieu Fradelizi ◽  
Olivier Guédon ◽  
Joseph Lehec ◽  
Carsten Schütt ◽  
...  
Keyword(s):  
Surface Area ◽  
Affine Surface ◽  
Affine Surface Area ◽  
Entropy Inequalities
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