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Reference.org
Strictly convex space
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a strictly convex space is a <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a> (<i>X</i>, || ||) for which the closed unit <a href="/facts/Ball_(mathematics)/f9vpMVMp">ball</a> is a strictly <a href="/facts/Convex_set/vdAuJRJl">convex set</a>. Put another way, a strictly convex space is one for which, given any two distinct points <i>x</i> and <i>y</i> on the <a href="/facts/Unit_sphere/5UeoTcug">unit sphere</a> ∂<i>B</i> (i.e. the <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> of the unit ball <i>B</i> of <i>X</i>), the segment joining <i>x</i> and <i>y</i> meets ∂<i>B</i> <i>only</i> at <i>x</i> and <i>y</i>. Strict convexity is somewhere between an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> (all inner product spaces being strictly convex) and a general <a href="/facts/Normed_space/rmDljfyE">normed space</a> in terms of structure. It also guarantees the uniqueness of a best approximation to an element in <i>X</i> (strictly convex) out of a convex subspace <i>Y</i>, provided that such an approximation exists. </p><p>If the normed space <i>X</i> is <a href="/facts/Banach_space/sDEIk7EC">complete</a> and satisfies the slightly stronger property of being <a href="/facts/Uniformly_convex_space/H9gGXdy7">uniformly convex</a> (which implies strict convexity), then it is also reflexive by <a href="/facts/Milman%25E2%2580%2593Pettis_theorem/nRFUDnwB">Milman–Pettis theorem</a>. </p>