Euclidean projection on a set
An Euclidean projection of a point 
on a set 
is a point that achieves the smallest Euclidean distance from 
to the set. That is, it is any solution to the optimization problem
When the set 
is convex, there is a unique solution to the above problem. In particular, the projection on an affine subspace is unique.
Example: assume that is the hyperplane
The projection problem reads as a linearly constrained least-squares problem, of particularly simple form:
The projection of 
on turns out to be aligned with the coefficient vector 
. Indeed, components of 
orthogonal to 
don't appear in the constraint, and only increase the objective value. Setting 
in the equation defining the hyperplane and solving for the scalar 
we obtain 
, so that the projection is 
.
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凸集（Convex Set）：
Definitions
A subset 
of 
is said to be convex if and only if it contains the line segment between any two points in it:
![forall : x_1, x_2 in mathbf{C}, ;; forall : lambda in [0,1] ~:~ lambda x_1 + (1-lambda) x_2 in mathbf{C}.](http://inst.eecs.berkeley.edu/%7Eee127a/book/login/eqs/2011859315932702027-130.png)
Subspaces and affine sets, such as lines, planes and higher-dimensional ‘‘flat’’ sets, are obviously convex, as they contain the entire line passing through any two points, not just the line segment. That is, there is no restriction on the scalar 
anymore in the above condition.
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当且仅当上的子集包含任意二点直接的线段满足：
Examples:
A set is said to be a convex cone if it is convex, and has the property that if 
, then 
for every 
.
