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{{Math theorem
| name = {{visible anchor|Pryce's
| math_statement = Suppose <math>p : X \to \R</math> is a sublinear functional on a vector space <math>X</math> and that <math>C \subseteq X</math> is a non-empty convex subset.
If <math>x \in X</math> is a vector and <math>r_0, R > 0</math> are positive real numbers such that
<math display=block>p(x) + r_0 R ~<~ \inf_{c \in C} p\left(x + r_0 c\right)</math>
then for every positive real <math>\mathbf{t} > 0</math> there exists some <math>\mathbf{c_0} \in C</math> such that
<math display=block>p\left(x + r_0 \mathbf{c_0}\right) + \mathbf{t} R ~<~ \inf_{c \in C} p\left(x + r_0 \mathbf{c_0} + \mathbf{t} c\right).</math>
}}
Adding <math>\mathbf{t} R</math> to both sides of the hypothesis <math display=inline>p(x) + r_0 R \,<\, \inf_{} p\left(x + r_0 C\right)</math> and combining that with the conclusion gives
<math display=block>p(x) + r_0 R + \mathbf{t} R ~<~ \inf_{} p\left(x + r_0 C\right) + \mathbf{t} R ~\leq~ p\left(x + r_0 \mathbf{c_0}\right) + \mathbf{t} R ~<~ \inf_{} p\left(x + r_0 \mathbf{c_0} + \mathbf{t} C\right)</math>
which yields many more inequalities, including, for instance,
<math display=block>p(x) + r_0 R + \mathbf{t} R ~<~ p\left(x + r_0 \mathbf{c_0}\right) + \mathbf{t} R ~<~ p\left(x + r_0 \mathbf{c_0} + \mathbf{t} \mathbf{c_0}\right)</math>
in which an expression on one side of a strict inequality <math>\,<\,</math> can be obtained from the other by replacing the symbol <math>R</math> with <math>\mathbf{c_0}</math> (or vice versa) and moving the closing parenthesis to the right (or left) of the adjacent summand (the symbols <math>p, x, r_0,</math> and <math>x</math> remain fixed in their places).
===Associated seminorm===
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