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I have taken the following from the review of the following paper "Schauder's conjecture on convex metric spaces" written in 2010 :

One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is:

Schauder's Conjecture. Let $E$ be a compact convex subset in a topological vector space. Then any continuous mapping $f:E\to E$ has a fixed point.

In this paper, the authors prove that it holds for convex metric spaces and consequently compact convex subsets of a $CAT(0)$ space have the fixed point property for continuous mappings.

So it seems that the problem in its general form is still open.

I have taken the following from the review of the following paper "Schauder's conjecture on convex metric spaces" written in 2010 :

One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is:

Schauder's Conjecture. Let $E$ be a compact convex subset in a topological vector space. Then any continuous mapping $f:E\to E$ has a fixed point.

In this paper, the authors prove that it holds for convex metric spaces and consequently compact convex subsets of a $CAT(0)$ space have the fixed point property for continuous mappings.

So it seems that the problem in its general form is still open.

I have taken the following from the review of the following paper "Schauder's conjecture on convex metric spaces" written in 2010 :

One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is:

 

Schauder's Conjecture. Let $E$ be a compact convex subset in a topological vector space. Then any continuous mapping $f:E\to E$ has a fixed point.

 

In this paper, the authors prove that it holds for convex metric spaces and consequently compact convex subsets of a $CAT(0)$ space have the fixed point property for continuous mappings.

So it seems that the problem in its general form is still open.

I have taken the following from the review of the following paper "Schauder's conjecture on convex metric spaces" written in 2010 :

One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is:

Schauder's Conjecture. Let $E$ be a compact convex subset in a topological vector space. Then any continuous mapping $f:E\to E$ has a fixed point.

In this paper, the authors prove that it holds for convex metric spaces and consequently compact convex subsets of a $CAT(0)$ space have the fixed point property for continuous mappings.

So it seems that the problem in its general form is still open.

I have taken the following from the review of the following paper "Schuader's conjecture on convex metric spaces" written in 2010 :

One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is:

Schauder's Conjecture. Let $E$ be a compact convex subset in a topological vector space. Then any continuous mapping $f:E\to E$ has a fixed point.

In this paper, the authors prove that it holds for convex metric spaces and consequently compact convex subsets of a $CAT(0)$ space have the fixed point property for continuous mappings.

So it seems that the problem in its general form is still open.

I have taken the following from the review of the following paper "Schauder's conjecture on convex metric spaces" written in 2010 :

One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is:

Schauder's Conjecture. Let $E$ be a compact convex subset in a topological vector space. Then any continuous mapping $f:E\to E$ has a fixed point.

In this paper, the authors prove that it holds for convex metric spaces and consequently compact convex subsets of a $CAT(0)$ space have the fixed point property for continuous mappings.

So it seems that the problem in its general form is still open.