Convex Curves in $\RP^n$
S.S.Anisov

Convex curves, that is, curves in $\RP^n$ intersecting any hyperplane at
most n$ times, arise between singularity theory, projective geometry,
differential geometry, and theory of Chebyshev systems of functions.
Author's research was stimulated by V.~I.~Arnold's paper~\cite4, which is
devoted to the following theorem: if $\G$ is a curve in $\RP^{n+1}$ such
that its projection $\g=p_A(\G)$ (where $A\notin\G$) is a convex curve in
$\RP^n$, then there are at least $n 2$ different flattening points on~$\G$.
(Recently A.~O.~Viro proved the converse theorem~\cite9: for a nonconvex
curve $\g\subset\RP^n$ there exists its lifting with less than $n+2$
flattening points.)

Many questions and conjectures concerning convex curves are stated in
~\cite{4,~5}. This paper contains answers for most of them. In \S3 we prove
that any convex curve $\g\subset\RP^n$ is a projection $p_A(\G)$ of some
convex curve $\G\subset\RP^{n+1}$ passing through $A$; here $A$ is an
arbitrary point in $\RP^{n+1}\setminus\RP^n$. This enables us to find the
homotopy type of the space of convex curves in $\RP^n$ (which is a subset of
$C^n(S^1,\RP^n)$): it is homotopy equivalent to the orthogonal group
$\SO(n+1,\R)$ or to the group $\PSO(n+1,\R)$, depending on the parity of $n$
(see \S4; before M.~Shapiro has proved~\cite{19} that this space is
connected). In \S7 we prove that for a convex curve in even-dimensional
Euclidean space there exists a subspace of codimension~2 such that the
projection of the curve on this subspace also is convex. To this end, it is
necessary to study the structure of the sets $M_k(\G)$ of the hyperplanes
that intersect $\G$ exactly $k$ times (see \S6). As a by-product, this
implies that any Chebyshev system of functions on the circle contains (up to
linear transformation, normalization, and reparametrization) the functions
$\sin\f$ and $\cos\f$ simultaneously. In \S5 we consider various
generalizations of the concept of convexity. A more detailed review of
principal results can be found in \S2; the following sections contain the
proofs. Definitions are collected in \S1, which also contains two isolated
theorems that are not used below: Theorem~4 describes the convex hull of a
convex curve, and Theorem~5 shows that a convex curve cannot be longer than
$n\pi$.