# Outer Normal Transforms of Convex Polytopes

Outer Normal Transforms of Convex Polytopes For a d-dimensional convex polytope $$P\subset {\mathbb {R}}^d$$ P ⊂ R d , the outer normal unit vectors of its facets span another d-dimensional polytope, which is called the outer normal transform of P and denoted by $$P^*$$ P ∗ . It seems that this transform, which clearly differs from the usual polarity transform, was not seriously investigated until now. We derive results about polytopes having natural properties with respect to this transform, like self-duality or equality for the composition of it. Among other things we prove that if P is a convex polytope inscribed in the unit sphere with the property that the circumradii of its facets are all equal, then $$(P^*)^*$$ ( P ∗ ) ∗ coincides with P. We also prove that the converse is true when P or $$P^*$$ P ∗ has at most 2d vertices. For the three-dimensional case, we characterize the family of those tetrahedra T such that T and $$T^*$$ T ∗ are congruent. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# Outer Normal Transforms of Convex Polytopes

, Volume 72 (2) – Oct 26, 2016
17 pages

/lp/springer_journal/outer-normal-transforms-of-convex-polytopes-pe0ZqyoxNl
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-016-0617-9
Publisher site
See Article on Publisher Site

### Abstract

For a d-dimensional convex polytope $$P\subset {\mathbb {R}}^d$$ P ⊂ R d , the outer normal unit vectors of its facets span another d-dimensional polytope, which is called the outer normal transform of P and denoted by $$P^*$$ P ∗ . It seems that this transform, which clearly differs from the usual polarity transform, was not seriously investigated until now. We derive results about polytopes having natural properties with respect to this transform, like self-duality or equality for the composition of it. Among other things we prove that if P is a convex polytope inscribed in the unit sphere with the property that the circumradii of its facets are all equal, then $$(P^*)^*$$ ( P ∗ ) ∗ coincides with P. We also prove that the converse is true when P or $$P^*$$ P ∗ has at most 2d vertices. For the three-dimensional case, we characterize the family of those tetrahedra T such that T and $$T^*$$ T ∗ are congruent.

### Journal

Results in MathematicsSpringer Journals

Published: Oct 26, 2016

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations

Abstract access only

18 million full-text articles

Print

20 pages / month

PDF Discount

20% off