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.The 'if' part is trivial. Conversely, assume now that X {displaystyle X} is uniformly convex and that x , y {displaystyle x,y} are as in the statement, for some fixed 0 < ϵ ≤ 2 {displaystyle 0<epsilon leq 2} . Let δ 1 ≤ 1 {displaystyle delta _{1}leq 1} be the value of δ {displaystyle delta } corresponding to ϵ 3 {displaystyle {frac {epsilon }{3}}} in the definition of uniform convexity. We will show that ‖ x + y 2 ‖ ≤ 1 − δ {displaystyle left|{frac {x+y}{2}} ight|leq 1-delta } , with δ = min { ϵ 6 , δ 1 3 } {displaystyle delta =min left{{frac {epsilon }{6}},{frac {delta _{1}}{3}} ight}} . 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. A uniformly convex space is a normed vector space so that, for every 0 < ϵ ≤ 2 {displaystyle 0<epsilon leq 2} there is some δ > 0 {displaystyle delta >0} so that for any two vectors with ‖ x ‖ = 1 {displaystyle |x|=1} and ‖ y ‖ = 1 , {displaystyle |y|=1,} the condition