In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a strictly convex space is a
normed vector space
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The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
(''X'', , , , , ) for which the closed unit
ball
A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...
is a strictly
convex set
In geometry, a set of points is convex if it contains every line segment between two points in the set.
For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
. 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 a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
∂''B'' (i.e. the
boundary 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, ofte ...
(all inner product spaces being strictly convex) and a general
normed space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
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 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
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
(''X'', , , , , ) is strictly convex if and only if ''x'' ≠ ''y'' and , , ''x'' , , = , , ''y'' , , = 1 together imply that , , ''x'' + ''y'' , , < 2.
* A
normed vector space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
(''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
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
(''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
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
(''X'', , , , , ) is strictly convex
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the
modulus of convexity ''δ'' for (''X'', , , , , ) satisfies ''δ''(2) = 1.
See also
*
Uniformly convex space
*
Modulus and characteristic of convexity
References
*
Convex analysis
Normed spaces
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