In
mathematics, a strictly convex space is a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
(''X'', , , , , ) for which the closed unit
ball is a strictly
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
. Put another way, a strictly convex space is one for which, given any two distinct points ''x'' and ''y'' on the
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
∂''B'' (i.e. the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
of the unit ball ''B'' of ''X''), the segment joining ''x'' and ''y'' meets ∂''B'' ''only'' at ''x'' and ''y''. Strict convexity is somewhere between an
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
(all inner product spaces being strictly convex) and a general
normed space in terms of structure. It also guarantees the uniqueness of a best approximation to an element in ''X'' (strictly convex) out of a convex subspace ''Y'', provided that such an approximation exists.
If the normed space ''X'' is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and satisfies the slightly stronger property of being
uniformly convex (which implies strict convexity), then it is also reflexive by
Milman-Pettis theorem.
Properties
The following properties are equivalent to strict convexity.
* A
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
(''X'', , , , , ) is strictly convex if and only if ''x'' ≠ ''y'' and , , ''x'' , , = , , ''y'' , , = 1 together imply that , , ''x'' + ''y'' , , < 2.
* A
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
(''X'', , , , , ) is strictly convex if and only if ''x'' ≠ ''y'' and , , ''x'' , , = , , ''y'' , , = 1 together imply that , , ''αx'' + (1 − ''α'')''y'' , , < 1 for all 0 < ''α'' < 1.
* A
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
(''X'', , , , , ) is strictly convex if and only if ''x'' ≠ ''0'' and ''y'' ≠ ''0'' and , , ''x'' + ''y'' , , = , , ''x'' , , + , , ''y'' , , together imply that ''x'' = ''cy'' for some constant ''c > 0'';
* A
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
(''X'', , , , , ) is strictly convex
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
the
modulus of convexity ''δ'' for (''X'', , , , , ) satisfies ''δ''(2) = 1.
See also
*
Uniformly convex space In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936.
Definition
A uniformly convex space is a ...
*
Modulus and characteristic of convexity In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ''ε''-''δ'' definition of uni ...
References
*
{{Functional analysis
Convex analysis
Normed spaces