Question 13196: The Norm Triangle Inequality Supplies the Metric Triangle Bound

Original question and givens

Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[8, 3], y=[-2, -8], z=[3, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).

• G001 — Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[8, 3], y=[-2, -8], z=[3, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-norm-induced-metric

Step-by-step derivation

x=[8, 3];y=[-2, -8];z=[3, -2];same norm=l1

x-z=(x-y)+(y-z);universal norm triangle gives the metric triangle bound

x to z difference coordinate0 := sub(8, 3)

x to z difference coordinate0 = 5

x to z difference coordinate1 := sub(3, -2)

x to z difference coordinate1 = 5

x to z:norm=l1;left=[8, 3];right=[3, -2]

signed left-right=[5, 5];use same named norm

x to z:coordinate=0;signed value=5

absolute value=5

x to z:coordinate=1;signed value=5

absolute value=5

x to z l1 sum := sum(5, 5)

x to z l1 sum = 10

x to y difference coordinate0 := sub(8, -2)

x to y difference coordinate0 = 10

x to y difference coordinate1 := sub(3, -8)

x to y difference coordinate1 = 11

x to y:norm=l1;left=[8, 3];right=[-2, -8]

signed left-right=[10, 11];use same named norm

x to y:coordinate=0;signed value=10

absolute value=10

x to y:coordinate=1;signed value=11

absolute value=11

x to y l1 sum := sum(10, 11)

x to y l1 sum = 21

y to z difference coordinate0 := sub(-2, 3)

y to z difference coordinate0 = -5

y to z difference coordinate1 := sub(-8, -2)

y to z difference coordinate1 = -6

y to z:norm=l1;left=[-2, -8];right=[3, -2]

signed left-right=[-5, -6];use same named norm

y to z:coordinate=0;signed value=-5

absolute value=5

y to z:coordinate=1;signed value=-6

absolute value=6

y to z l1 sum := sum(5, 6)

y to z l1 sum = 11

triangle right hand sum := sum(21, 11)

triangle right hand sum = 32

Reverse checks

• S020 · premise_and_result_audit — Recompute triangle right hand sum with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute y to z l1 sum with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute y to z difference coordinate1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute y to z difference coordinate0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute x to y l1 sum with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute x to y difference coordinate1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute x to y difference coordinate0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute x to z l1 sum with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute x to z difference coordinate1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute x to z difference coordinate0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(l1, 10, 21, 11, 32, Yes, norm_triangle_inequality)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.