2023
DOI: 10.1112/mtk.12191
|View full text |Cite
|
Sign up to set email alerts
|

Isoperimetric problems for zonotopes

Abstract: Shephard (Canad. J. Math. 26 (1974), 302-321) proved a decomposition theorem for zonotopes yielding a simple formula for their volume. In this note, we prove a generalization of this theorem yielding similar formulae for their intrinsic volumes. We use this result to investigate geometric extremum problems for zonotopes generated by a given number of segments. In particular, we solve isoperimetric problems for 𝑑-dimensional zonotopes generated by 𝑑 or 𝑑 + 1 segments, and give asymptotic estimates for the so… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
1
1
1

Citation Types

0
3
0

Year Published

2023
2023
2023
2023

Publication Types

Select...
2

Relationship

1
1

Authors

Journals

citations
Cited by 2 publications
(3 citation statements)
references
References 46 publications
0
3
0
Order By: Relevance
“…We note that the statement of Theorem 7 for simplices can be found in [13]. Theorem For any 1kd1$1 \leqslant k \leqslant d-1$, among simplices in double-struckEd$\mathbb {E}^d$ with unit volume, the regular ones have minimal total k ‐content.…”
Section: Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…We note that the statement of Theorem 7 for simplices can be found in [13]. Theorem For any 1kd1$1 \leqslant k \leqslant d-1$, among simplices in double-struckEd$\mathbb {E}^d$ with unit volume, the regular ones have minimal total k ‐content.…”
Section: Resultsmentioning
confidence: 99%
“…Lemma 2 explores the properties of the so‐called linear parameter systems or shadow systems , introduced by Rogers and Shephard in [20]. Lemma 2 can be found in [13], and hence, we omit its proof. Lemma Let p1,,pk,vEd$p_1, \ldots , p_k, v \in \mathbb {E}^d$ and λ1,,λkdouble-struckR$\lambda _1, \ldots , \lambda _k \in \mathbb {R}$.…”
Section: Preliminariesmentioning
confidence: 99%
“…This problem has been studied by a variety of authors (see for instance [7][8][9][10]16,18]). Of particular interest is the case when Z = B d+1 is the Euclidean unit ball.…”
Section: Introductionmentioning
confidence: 99%