CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Nearest-point maps to convex sets need not be linear

Statement refuted

The nearest-point map $P_{C}$ of a nonempty closed convex set $C$ in a Hilbert space is linear.

Facts & Assumptions

[A1]

$R$ with the pairing $(s, t) \mapsto s t$ is a real inner-product space whose induced norm is the absolute value (Real and complex inner-product spaces and their induced length, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms), and it is complete, hence a Hilbert space (Complete metric space: every Cauchy sequence converges in the space, Banach space, Hilbert space).

[A2]

A set $C$ is convex when $(1 - t) u + t v \in C$ for all $u, v \in C$ and $0 \leq t \leq 1$ , and the nearest point of a nonempty closed convex subset of a Hilbert space is the point minimising the distance (Convex sets and continuous real-hyperplane separation in a normed space, Hilbert space).

Counterexample

technique · direct

Given: The Hilbert space $R$ of [A1] and the closed convex set $C = [0, \infty)$ .

1.1A1A2

$C$ is closed and convex, and is nonempty with $0 \in C$ .

2.1step 1.1A1A2algebra

For $t \geq 0$ the point $c = t$ lies in $C$ with $∣ t - c ∣ = 0$ , so $P_{C} (t) = t$ ; for $t < 0$ and any $c \in C$ one has $∣ t - c ∣ = c - t \geq - t = ∣ t - 0 ∣$ with equality exactly at $c = 0$ , so $P_{C} (t) = 0$ .

3.1step 2.1algebra

Hence $P_{C} (t) = \max {t, 0}$ ; this map is not additive, since $P_{C} (1) + P_{C} (- 1) = 1 + 0 = 1 \neq 0 = P_{C} (0)$ , and it is not homogeneous either, since $P_{C} (- 1) = 0 \neq - 1 = - P_{C} (1)$ .

4.1step 3.1∎

Therefore the nearest-point map of a nonempty closed convex set in a Hilbert space need not be linear, so the statement refuted is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 1.44, pp.40–41 (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 178 (standard reference, not scraped)