You mean proximity in Euclidean space vs that in DTOA space.

'Proximity' means 'distance'.

Euclidean space is just a space defined over the real numbers (meaning that all coordinates are specified by a real number), and has a positive definite symmetric bilinear form (for example, the scalar product). Euclidean space can be n-dimensional, but in your paper, it's just 2-dimensional (denoted R^2). As stated in your paper, the distance between two points P=(x,y) and Q=(u,v) in R^2 is d(P,Q)=sqrt((x-u)^2+(y-v)^2).

DTOA space (Difference of Time-of-Arrival space) is (in this case) what's called a 'dual vector space' of R^2. It's a fairly complicated concept, but you can just think of it as another vector space for which all vectors in R^2 can be mapped to a unique vector in DTOA space. The 'proximity' (distance) then on DTOA space, is again just the distance between two points P' and Q' in DTOA space, d(P', Q').

The paper then goes on to give some relationships between these two.

'Proximity' means 'distance'.

Euclidean space is just a space defined over the real numbers (meaning that all coordinates are specified by a real number), and has a positive definite symmetric bilinear form (for example, the scalar product). Euclidean space can be n-dimensional, but in your paper, it's just 2-dimensional (denoted R^2). As stated in your paper, the distance between two points P=(x,y) and Q=(u,v) in R^2 is d(P,Q)=sqrt((x-u)^2+(y-v)^

DTOA space (Difference of Time-of-Arrival space) is (in this case) what's called a 'dual vector space' of R^2. It's a fairly complicated concept, but you can just think of it as another vector space for which all vectors in R^2 can be mapped to a unique vector in DTOA space. The 'proximity' (distance) then on DTOA space, is again just the distance between two points P' and Q' in DTOA space, d(P', Q').

The paper then goes on to give some relationships between these two.