Gruber P. Convex and Discrete Geometry

16 Convex Functions
Minkowski’s Inequality for Sums
Minkowski’s inequality for sums or the triangle inequality for the p-norm is as follows.
Corollary 1.6. Let x1, y1,...,xn, yn ≥ 0 and p ≥ 1. Then
�
(x1 + y1) p +···+(xn + yn) p� 1 �
p p
≤ x1 +···+x p� 1 �
p p
n + y1 +···+y p� 1
p
n .
Proof. For p = 1 this inequality holds trivially. Assume now that p > 1. Let q > 1
such that 1 p + 1 q = 1. Then (p − 1)q = p and thus
(x1 + y1) p +···+(xn + yn) p
= x1(x1 + y1) p−1 +···+xn(xn + yn) p−1
+ y1(x1 + y1) p−1 +···+yn(xn + yn) p−1
≤ � x p
1 +···+x p� 1 �
p
n (x1 + y1) p +···+(xn + yn) p� 1 q
+ � y p
1 +···+y p� 1 �
p
n (x1 + y1) p +···+(xn + yn) p� 1 q
= � (x1 + y1) p +···+(xn + yn) p� 1 q � (x p
1 +···+x p n ) 1 p + (y p
1 +···+y p n ) 1 p �
by Hölder’s inequality. ⊓⊔
Minkowski’s Inequality for Integrals
A similar argument gives our last result.
Corollary 1.7. Let f, g : I → R be non-negative and integrable and let p ≥ 1.
Then
� �
( f + g) p � 1 �
p
dx ≤
�
f p � 1 �
p
dx +
�
g p � 1
p
dx .
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1.5 Bohr and Mollerup’s Characterization of Ɣ
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A given functional equation may have many solutions. To single out interesting special
solutions, additional conditions must be imposed, for example continuity, measurability,
boundedness or convexity conditions. In the case of Cauchy’s functional
equation of 1821,
many such results are known.
f (x + y) = f (x) + f (y) for x, y ∈ R,
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