Timeline for What are the injective embeddings of R^d into the cone of (semi-) positive definite matrices of dimension d?

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Sep 12 at 15:08	comment	added	Ali Taghavi		More generally is every compact convex set in $\mathbb{R}^n$ homeomorphic to a $k$ dimensional disk for some $k\leq n$?
Sep 12 at 15:05	comment	added	Ali Taghavi		@RobertBryant What can be said about the topology of space of all semi positive matrix of size $d$ whose matrix norm is $\leq$ 1? It is compact and convex so is it homeomorphic to a disk?
Sep 3 at 12:35	comment	added	Robert Bryant		Since $\exp:S_d(\mathbb{R})\to S^+_d(\mathbb{R}$) is a diffeomorphism, where $S_d(\mathbb{R})$ is the vector space of symmetric $d$-by-$d$ matrices with real entries (whose dimension is $\tfrac12d(d{+}1)$), and $S^+_d(\mathbb{R})\subset S_d(\mathbb{R})$ is the open cone of positive definite symmetric $d$-by-$d$ matrices, any injection $f:\mathbb{R}^d\to S^+_d(\mathbb{R})$ is uniquely of the form $f = \exp\circ\phi$, where $\phi:\mathbb{R}^d\to S_d(\mathbb{R})$ is an injective map, so you are reduced to 'characterizing' the injective maps $\phi$ into the vector space $S_d(\mathbb{R})$.
Sep 1 at 23:53	comment	added	Yemon Choi		You need to specify some extra restrictions on your functions (such as continuity, or linearity) - but until you edit your question it is not clear to readers what tacit conditions you are imposing.
Aug 31 at 16:05	review	Close votes
Sep 19 at 3:01
S Aug 31 at 14:29	review	First questions
Aug 31 at 15:46
S Aug 31 at 14:29	history	asked	Drmanifold	CC BY-SA 4.0