Volume estimates for sections of certain convex bodies |
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Authors: | Patryk Brzezinski |
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Institution: | 1. + 2. 49 3. 431 4. 8801155+49 431 8804091;5. Mathematisches Seminar, Christian‐Albrechts‐Universit?t zu Kiel, , 24118 Kiel, Germany |
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Abstract: | This paper deals with volume estimates for hyperplane sections of the simplex and for m‐codimensional sections of powers of m‐dimensional Euclidean balls. In the first part we consider sections through the centroid of the n‐dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m‐codimensional sections “perpendicular” to of unit balls in the space for all . We give volume formulas and use them to show that the normal vector (1, 0, …, 0) yields the minimal volume. Furthermore we give an upper bound for the ‐dimensional volumes for natural numbers . This bound is asymptotically attained for the normal vector as . |
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Keywords: | Regular simplex cube slicing probabilistic methods volume of sections Bessel functions 52A20 52A21 46B07 46B09 33C10 |
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