In a linear space over the field of real numbers, the segment between the points A and B is the following set:
{ (1-u) A + u B | u Î [0,1] }
In this, [0,1] denotes the interval consisting of all real numbers between 0 and 1 (both extremities included).
A subset of a real linear space is said to be convex if it contains all the segments between every pair of its points.
Convexity is not defined for linear spaces whose scalar fields are not endowed with a total ordering relation (e.g., p-adic numbers).
A real-valued function f of a vectorial variable is said to be convex when it's defined over a convex open set V and the following holds:
" A Î V ,
" B Î V ,
" u Î ]0,1[ ,
f ( (1-u) A + u B )
≤
(1-u) f ( A ) + u f ( B )
f is strictly convex if that inequality is always strict for distinct A & B.
Aleksandrov's theorem states that, almost everywhere, all second-order partial derivatives of a convex function exist and the Hessian matrix formed by them is positive semi-definite.
For the Euclidean plane, that was first proved in 1936 by Herbert Busemann (1905-1994) and William Feller (1906-1970): Krümmungseigenschaften konvexer Flächen, Acta Mathematica, 66, pp. 1-47.
The theorem was extended in 1939 by Alexandrov to any convex real-valued function f of an n-dimensional Euclidean variable.
Wikipedia : Alexandrov's theorem | Aleksandr Danilovich Aleksandrov (1912-1999)
Let E be a normed vector space whose unit ball is:
B = { z Î E | || z || ≤ 1 }
E is a uniformly convex space if and only if the following is true:
" e > 0, $ d > 0, "xÎB, "yÎB, { || x - y || ≤ e } Þ { || ½ (x+y) || ≤ 1-d }
Wikipedia : Uniformly convex space | Milman-Pettis theorem (1938 & 1939)
Technically, the width of a convex shape depends on the direction with respect to which it is measured: It's the least possible distance of two parallel planes perpendicular to that direction and surrounding the body. It's what a sliding calipers measures.
The height of the shape can be given either of the following definitions:
The former definition is called absolute, the latter is dubbed relative.
In either case, the length of the body is then defined as the largest width along an horizontal direction (a direction is said to be horizontal when it's perpendicular to the aforementioned vertical direction).
Either way, the aspect ratio of a body is its length to height ratio.
Either flavor of aspect ratio can be a convenient parameter to use when discussing the geometry of a family of objects. It makes little sense for nonconvex things, except by considering their convex hulls. An absolute aspect ratio is never less than unity. A relative aspect ratio can be.
The aspect ratio normally quoted for a rectangle is, in fact, its relative aspect ratio when the smaller side is vertical. It's an exercise in elementary trigonometry (left to the reader) to show that the absolute aspect ratio differs from that by a factor of 2 cos2 q , where q is the angle between the longer side and the diagonal (that's between 0° and 45°). In other words:
Usual aspect ratio = 1 / tg q Absolute aspect ratio = 1 / sin 2q
Curiously, the absolute aspect ratio of a very shallow rectangle is thus about half its aspect ratio in the usual sense of the term!
The diameter of a body is simply its largest width. For example, the diameter of a rectangle is equal to its diagonal.
Playing Cards | Aspect ratio (7:11 video) by Tony Northrup (2017年12月19日).
The unit ball associated with a norm is defined as the set of all vectors whose norm is less than or equal to 1. The basic properties of a norm always make this a closed convex set symmetric about the origin (i.e., it contains -V if it contains V ) and containing more than 0 itself (except in the trivial case of the space {0} of dimension zero).
Conversely (Minkowski) any such set B uniquely specifies a norm of which it is the unit ball, the norm of a vector V being defined as:
|| V || = inf { |x| | V Î x B }
The above definition of a norm from a convex, origin-symmetric, closed body B is called Minkowski's functional.
The notation x B was introduced by Minkowski himself. It denotes the set of all vectors that are equal to the scalar x multiplied into some element of B. Likewise, Minkowski defined the sum A+B of two sets of vectors as the set of all vectors that can be obtained by adding together an element of A and an element of B. Similarly, any well-defined operation on the elements of sets induces a natural extension of that operation to sets themselves, which is sometimes called a Minkowski operation on sets (the most common is Minkowski addition, which has nice convexity properties).
Video :
In a "circle", the circumference-to-diameter ratio is between 3 and 4
(PBS infinite series, 2017年01月05日).
Wikipedia :
Minkowski functional
The intersection of any family of convex sets is itself convex. (HINT: If two points are in that intersection, so is the segment between them.)
Therefore, the intersection of all convex sets that contain a set S of vectors is a convex set containing S. It's clearly the smallest such set. It's called the convex hull of S and it's usually denoted Conv (S).
The convex hull of a sum of sets is the sum of their convex hulls:
Conv ( S1 + S2 ) = Conv ( S1 ) + Conv ( S2 )
A convex hull is not necessarily closed. (Consider, for example, an open halfspace together with a single point on its boundary.)
The closure of Conv (S) is the convex hull of the closure of S. It's called the closed convex hull of S :
Conv ( S ) = Conv ( S )
As discussed next, a closed convex set is the intersection of all [closed] halfspaces that contain it. The closure of the convex hull of S (or, equivalently, the convex hull of the closure of S) is the intersection of all halfspaces that contain S.
Wikipedia : Shapley-Folkman lemma : The [Minkowski] sum of many sets is nearly convex.
An hyperplane separates space into three disjoint regions; itself and two open halfspaces. A closed halfspace is obtained as the union of the hyperplane with either of the two open halfspaces it borders.
When we say that any closed convex set is the intersection of halfspaces, we're normally thinking about closed ones. It's "more economical" to do so, but it's not strictly necessary (since a closed halfspace containing S is clearly the intersection of infinitely many open halfspaces).
The converse proposition only holds for closed halfspaces, though. An intersection of any family of closed sets is guaranteed to be closed (an infinite intersection of open sets could be open, closed or neither).
{ closed convex sets } = { intersections of closed halfspaces }
This proposition is the geometric Hahn-Banach theorem.
Convex Optimization by Stephen P. Boyd & Lieven Vandenberghe.
In Euclidean space (i.e., real linear space endowed with a positive-definite scalar product) a linear hyperplane can be defined as the set of all vectors orthogonal to a prescribed nonzero vector. An affine hyperplane (or, simply, an hyperplane) is obtained by adding to some point every vector from such a linear hyperplane.
An hyperplane which does not go through the origin can be characterized by an othogonal vector H pointing to it from the origin with a length equal to the inverse of the Euclidean distance from the origin. That hyperplane is the set of all vectors whose dot-product into H is equal to 1 :
{ V | V.H = 1 }
The hyperplane is the border of a closed half-plane containing the origin:
{ V | V.H ≤ 1 }
As stated in the previous section, we may always define any closed convex set C as the intersection of (possibly infinitely many) such closed half-spaces, namely:
C = { V | "HÎC' , V.H ≤ 1 }
All sets C' that yield C in this way have the same convex hull C* which is called the polar of C. The bodies C and C* are polars of each other :
C* = { V |
"HÎC ,
V.H ≤ 1 }
C = { V |
"HÎC* ,
V.H ≤ 1 }
If C and C* are polytopes (resulting from a finite C' ) then their networks of vertices, edges and faces are topological duals of each other.
[画像: Icosidodecahedron ] [画像: ] [画像: Rhombic triacontahedron ]
The above describes duality with respect to a sphere (or hypersphere) of unit radius centered at the origin. Any center and any radius could be used in practice.
If the two disjoint convex sets are neither open nor closed, the hyperplane at the border of the two halfspaces may intersect both convexes...
Consider, for example, the following bounded planar regions:
{ (x,y) | x2+y2 ≤ 1 and either y > 0 or [ y = 0 & x > 0 ] }
{ (x,y) | x2+y2 ≤ 1 and either y < 0 or [ y = 0 & x < 0 ] }
Each region is convex and the two are disjoint. Only one straight line (of equation y = 0) can be drawn "between" them, but it intersects both.
If both sets are closed, one of them can be contained in a closed halfspace which doesn't intersect the other. [ Conjecture. ]
If at least one of them is compact, then two disjoint closed convex sets can always be separated by an hyperplane which doesn't intersect either set (in fact, infinitely many such hyperplanes exist, in that case).
Two open disjoint convex sets can always be separated by at least one hyperplane that doesn't intersect either of them.
Hyperplane separation theorem
|
Hermann Minkowski (1864-1909)
MIT 18.409, by Jonathan Kelner :
Convex geometry |
Separating hyperplanes
The theorem doesn't apply for closed sets if neither is compact. Example:
Both regions are convex and closed, yet no straight line separates them. (HINT: To separate anything from the lower convex set (blue) a straight line must have zero slope and positive intercept.)
Separating hyperplane is perpendicular to separating axis.Let A and B be two disjoint convexes in the vector space E, such that A is compact and B is closed. Consider the following function f, defined for any point M of E :
f (M) = inf { d(M,x) | x Î B }
f is well-defined because any set of nonnegative reals has a lower bound. It is continuous.
[画像: Come back later, we're still working on this one... ]
