%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Hodgson, Craig D.; Rivin, Igor; Smith, Warren D.
%%
%% [Convex hyperbolic and convex polyhedra in the sphere]{A
%% characterization of convex hyperbolic polyhedra and of convex
%% polyhedra inscribed in the sphere
%%
%% We describe a characterization of convex polyhedra in $\h^3$ in terms of
%% their dihedral angles, developed by Rivin. We also describe some
%% geometric and combinatorial consequences of that theory. One of these
%% consequences is a combinatorial characterization of convex polyhedra
%% in $\E^3$ all of whose vertices lie on the unit sphere. That resolves
%% a problem posed by Jakob Steiner in 1832.
%%
%% publ: Bull. Amer. Math. Soc. (N.S.) 27(1992) no. 2
%% pp: 246-251
%% type: Research Announcement markup: amslatex file size: 28K
%%
%% copyright: American Math. Society copyright; see end of article
%%
%% Include files necessary for this article: bull-art.tex
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle{amsart}
\input{bull-art}
%[12pt]{article}
\bibliographystyle{numbered}
\def\descriptionlabel#1{\normalshape\rm\kern-.5em#1}
\newcommand{\mycirc}{\hbox{\raise.25ex\hbox{${\scriptstyle
\circ}$}}\ }
\newcommand{\ig}{\ell}
\newcommand{\ttot}{\tau_{\mbox{\scriptsize total}}}
\renewcommand{\varphi}{\wp}
\newcommand{\R}{\bold{R}} %added by cdh
\newcommand{\h}{\bold{H}} %added by cdh; latex already
%uses \H for something
\newcommand{\s}{\bold{S}} %added by cdh; latex already
%uses \S for something
\newcommand{\E}{\bold{E}} %added by cdh
\newcommand{\p}{{\cal P}}
\newcommand{\m}{{\cal M}}
\newcommand{\BE}{\begin{equation}}
\newcommand{\EE}{\end{equation}}
\newcommand{\BA}{\begin{eqnarray}}
\newcommand{\EA}{\end{eqnarray}}
\newcommand{\BC}[1]{\begin{figure}[ht]\vspace{0.5in}%
\caption #1}
\newcommand{\EC}{\end{figure}}
\newcommand{\thefig}{\thefigure}
\newcommand{\bd}[1]{{\bf #1}} % \bd{stuff} puts stuff in
%bold
%%\newcommand{\pf}[1]{{\smallskip\noindent\bf #1}} % for
%proof headings
\newcommand{\fn}[1]{{\tiny #1}}
\newcommand{\cp}[1]{{\sc #1}} % \cp{stuff} puts stuff in
%small caps
\newcommand{\ic}[1]{{\it #1}} % \ic{stuff} puts stuff in
%italics
\newcommand{\es}[1]{{\em #1}} % \es{stuff} puts stuff in
% italics/roman toggle
\newcommand{\ro}[1]{\mathrm{#1}} % \ro{stuff} puts stuff
%in roman
\newcommand{\Kext}{K_{\mbox{\scriptsize ext}}}
\newcommand{\QED}{\hfill $\Box$\smallskip}
\newcommand{\sphi}{$\bigcirc_\infty$} % sphere at
%hyperbolic infinity BROKEN!!
\newcommand{\ij}{\ro{InjRad}} % injectivity radius
\let\mcol=\multicolumn
\def\0{\kern.5em} % width of a digit
\newcommand{\po}{{\cal P}} %polar map (script P)
\newcommand{\poi}{{\cal P}^{- 1}} %inverse polar map
%(script P^-1)
%%\newcommand{\implies}{\Rightarrow}
\def\di{\mathop {\rm dist}\nolimits} %distance
\newcommand{\len}{\ell} %length
\newcommand{\dl}{\ro{dist}_L} %lipschitz dist
\newcommand{\dh}{\ro{dist}_H} %haussdorf dist
\def\arctanh{\mathop {\rm arctanh}\nolimits}
\def\arcsec{\mathop {\rm arcsec}\nolimits}
\def\sign{\mathop {\rm sign}\nolimits}
\newcommand{\degree}{^{\circ}}
\newcommand{\degrees}{^{\circ}}
\newcommand{\arccosh}{\ro{arccosh}}
\def\arccosh{\mathop {\rm arccosh}\nolimits}
\def\arcsinh{\mathop {\rm arcsinh}\nolimits}
\def\diam{\mathop {\rm diam}\nolimits}
\newcommand{\Bd}{{\partial }} %added by cdh
\newcommand{\ip}[3]{\langle #1 \rangle_{#2 , #3}} %inner
%product (new)
\newcommand{\nr}[3]{\| #1 \|_{#2 , #3}} %norm (new)
\newcommand{\FIG}[1]{\vspace{ #1 }} % puts #1 (which is
%``1.5in'' or
% something) blank space for pastein
\newcommand{\leb}[1]{} %for uncited labels
\newcommand{\lab}[1]{\label{#1}}
\newcommand{\rf}[1]{\ref{#1}}
\newcommand{\numb}[1]{(\ref{#1})}
%%\newcommand{\thm}[2]{{\noindent\begin{#1} #2 \end{#1}}}
\newcommand{\ie}[1]{{\index{#1}}}
\renewcommand{\Sp}{\bold{S}}
\newcommand{\mpf}[1]{\marginpar{\fn#1}}
%%\bibliographystyle{alpha}
\newtheorem{Theorem}{Theorem}
\newtheorem{Steinitz}{Theorem of Steinitz}
\renewcommand{\theSteinitz}{}
\theoremstyle{remark}
\newtheorem{Notes}{Notes}
\renewcommand{\theNotes}{}
\newtheorem{Note}{Note}
\renewcommand{\theNote}{}
\theoremstyle{definition}
\newtheorem{chRast}{Characterization $R^\ast$\defaultfont}
\renewcommand{\thechRast}{}
\newtheorem{chR}{Characterization $R$\defaultfont}
\renewcommand{\thechR}{}
\begin{document}
\def\currentvolume{27}
\def\currentissue{2}
\def\currentyear{1992}
\def\currentmonth{October}
\def\copyrightyear{1992}
\def\currentpages{246-251}
\title[Convex hyperbolic and convex polyhedra in the
sphere]{A
characterization of convex hyperbolic polyhedra and of
convex polyhedra inscribed in the sphere}\\endtitle
%%\footnote{1980 Mathematics Subject Classification (1985
%Revision):
\author[C. D. Hodgson]{Craig~D.~Hodgson}
\address{Mathematics Department, University of
Melbourne, Parkville, Victoria, Australia}
\author{Igor~Rivin}
\address{NEC Research Institute, Princeton, New Jersey 08540
and Mathematics Department, Princeton University,
Princeton, New Jersey 08540}
\author[W. D. Smith]{Warren D. Smith}
\address{NEC Research Institute, Princeton, New Jersey
08540}
\subjclass{Primary 52A55, 53C45, 51M20; Secondary 51M10,
05C10, 53C50}
\date{August 30, 1991}
\maketitle
\begin{abstract}
We describe a characterization of convex polyhedra
in $\h^3$ in
terms of their dihedral angles, developed by Rivin. We
also describe
some geometric and combinatorial consequences of that
theory. One of
these consequences is a combinatorial characterization of
convex
polyhedra in $\E^3$
all of whose vertices lie on the unit sphere. That
resolves a
problem posed by Jakob Steiner in 1832.
\end{abstract}
In 1832, Jakob Steiner in his book \cite{st:geom}
asked the following question:
\begin{quote}
In which cases does a convex polyhedron have a
(combinatorial)
equivalent which is inscribed in, or circumscribed about,
a sphere?
\end{quote}
This was the 77th of a list of 85 open problems posed by
Steiner,
of which only numbers 70, 76, and 77
were still open as of last year.
Apparently Ren\'e Descartes was also interested
in the problem (see \cite{de:solid}).
Several authors found families of
noninscribable polyhedral types, beginning with Steinitz
in 1927
(cf. \cite{gru:conp}); all of these families later were
subsumed by a
theorem of Dillencourt \cite{Dill90b}. In their 1991 book
\cite[problem B18]{CFG}, Croft, Falconer, and Guy
had the following to say:
\begin{quote}
It would of course be nice to characterize the polyhedra of
inscribable type, but as this may be over-optimistic, good
necessary,
or sufficient, conditions would be of interest.
\end{quote}
Here we announce a full answer to Steiner's question, in
the sense
that we produce a characterization of inscribable (or
circumscribable) polyhedra that has a number of pleasant
properties---it
can be checked in polynomial time and it yields a number of
combinatorial corollaries. First we note the following
well-known
characterization of convex polyhedra proved by Steinitz (cf.
\cite{gru:conp}).
%%\medskip\noindent
\begin{Steinitz}
A graph is the one-skeleton of a convex
polyhedron in $\E^3$ if and only if it is
a $3$-connected planar graph.
\end{Steinitz}
%%\noindent
\begin{Note}
A graph $G$ is {\em $k$-connected} if the complement of any
$k-1$ edges in $G$ is connected.
We will call graphs satisfying the criteria of Steinitz'
theorem {\em
polyhedral graphs}.
The answer to Steiner's question stems from the following
characterization of ideal convex polyhedra in hyperbolic
3-space $\h^3$.
(See \cite{th:gt3m, bear:grps} for the basics of hyperbolic
geometry.)
\end{Note}
%%\medskip\noindent
\begin{Theorem}
Let $P$ be a polyhedral graph with weights $w(e)$
assigned to the
edges. Let $P^*$ be the planar dual \RM(or Poincar\'e
dual\/\RM)
of $P$, where the edge $e^*$ dual to $e$ is assigned the
dual weight
$w^*(e^*)= \pi - w(e)$. Then $P$ can be realized as a convex
polyhedron in $\h^3$ with all vertices on the sphere at
infinity and
with dihedral angle $w(e)$ at every edge $e$ if and only
if the
following conditions hold\/\rom{:}
\begin{enumerate}
\item{} $0