**Properties of CAT( k) Spaces**

Let (*X*, *d*) be a CAT(*k*) space. Then the following properties hold:

- Given any two points
*x*,*y*∈*X*(with*d*(*x*,*y*) <*D*_{k}if*k*> 0), there is a unique geodesic segment that joins*x*to*y*; moreover, this segment varies continuously as a function of its endpoints.

- Every local geodesic in
*X*with length at most*D*_{k}is a geodesic.

- The
*d*-balls in*X*of radius less than ½*D*_{k}are (geodesically) convex.

- The
*d*-balls in*X*of radius less than*D*_{k}are contractible.

- Approximate midpoints are close to midpoints in the following sense: for every
*λ*<*D*_{k}and every*ε*> 0, there exists a*δ*=*δ*(*k*,*λ*,*ε*) > 0 such that, if*m*is the midpoint of a geodesic segment from*x*to*y*with*d*(*x*,*y*) ≤*λ*and

- then
*d*(*m*,*m*′) <*ε*.

- It follows from these properties that, for
*k*≤ 0, the universal cover of every CAT(*k*) space is contractible; in particular, the higher homotopy groups of such a space are trivial. As the example of the*n*-sphere**S***n*shows, there is, in general, no hope for a CAT(*k*) space to be contractible if*k*is strictly positive.

- An n-dimensional CAT(
*k*) space equipped with the n-dimensional Hausdorff measure satisfies the CD condition in the sense of Lott-Villani-Sturm.

Read more about this topic: CAT(k) Space

### Famous quotes containing the words spaces and/or properties:

“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of *spaces* of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”

—Blaise Pascal (1623–1662)

“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the *properties* of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”

—John Locke (1632–1704)