The first step of enriching the class of uniform measures on convex domains to that of non-negatively curved (""log-concave"") measures in

Euclidean space has been very successfully implemented in the last decades, leading to substantial progress in our understanding of volumetric properties of convex domains, mostly regarding concentration of linear functionals.

In particular, there is a nonlinear relation between the

Euclidean space and the Aitchison geometry, therefore making it inappropriate for standard statistical methods designed for unconstrained data to be applied directly to constrained compositional data.

So when the author says, "Whereas

Euclidean space is directly grasped by intuition, others of necessity are referable to the Euclidean notion of space for their intelligibility," this is wide of the mark.

Let M be an n-dimensional submanifold of an (n + m)-dimensional

Euclidean space [E.

While balls in

Euclidean space are rotation invariant, the same is not true in spaces endowed with other metrics.

Flattening it in the 2-dimesional

Euclidean space [R.

Multivector (3) associated with n-dimensional

Euclidean space can be represented by the [2.

1] be two curves in the three-dimensional

Euclidean space [E.

Moreover the minimal surfaces of translation of a higher dimensional

Euclidean space are obtained in [13] and of a semi-

Euclidean space are investigated in [12].

An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in

Euclidean space.

In 1963, Gahler introduced the notion of a 2-metric space, a real valued function of point-triples on a set X, whose abstract properties were suggested by the area function for a triangle determined by a triple in

Euclidean space.

Kant's claim about the necessity of

Euclidean space was supposedly an example of synthetic a priori "knowledge" (presumably about reality).

My reacting or circumscribing may be carried out in

Euclidean space or one of constant positive or of constant negative (or, for that matter, of variable) curvature.

Medical centers that are close on characteristics will also naturally be close in

Euclidean space.

Terras, Finite analogues of

Euclidean space, Journal of Computational and Applied Mathematics, 68 (1996), 221-238.