Abstract
As a substraction counterpart of the well-known p-sum of convex bodies, we introduce the notion of p-difference. We prove several properties of the p-difference, introducing also the notion of p-(inner) parallel bodies. We prove an analog of the concavity of the family of classical parallel bodies for the p-parallel ones, as well as the continuity of this new family, in its definition parameter. Further results on inner parallel bodies are extended to p-inner ones; for instance, we show that tangential bodies are characterized as the only convex bodies such that their p-inner parallel bodies are homothetic copies of them.
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Acknowledgments
The authors would like to sincerely thank M. A. Hernández Cifre for enlightening discussions.
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A. R. Martínez Fernández and J. Yepes Nicolás are supported by MINECO-FEDER project MTM2012-34037.
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Martínez Fernández, A.R., Saorín Gómez, E. & Yepes Nicolás, J. p-difference: a counterpart of Minkowski difference in the framework of the \(L_p\)-Brunn–Minkowski theory. RACSAM 110, 613–631 (2016). https://doi.org/10.1007/s13398-015-0253-3
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DOI: https://doi.org/10.1007/s13398-015-0253-3
Keywords
- p-difference
- p-inner parallel body
- Minkowski difference
- Relative inradius
- (p-)kernel
- (\(+_p\)-)concave family
- Tangential body