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  "body": "\n## Summary\n\nA normed space is strictly convex when its unit sphere contains no line segment.\nEquivalently, the midpoint of two distinct unit vectors has norm less than $1$.\nStrict convexity gives uniqueness results, such as uniqueness of the nearest point of a convex set and equality cases in the triangle inequality.\nIt is weaker than [[uniform convexity|BEAMC5011M]].\n\n## Explanations\n\n### L1 human\n\nLook at the boundary of the unit ball of a normed space.\nIn a strictly convex space this boundary is curved everywhere and has no flat pieces.\nThe Euclidean disc is strictly convex.\nA square is not, because its sides are flat segments.\n",
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