Question 13178: 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=[-10, -1], y=[-2, 10], z=[4, 0]. 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=[-10, -1], y=[-2, 10], z=[4, 0]. 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=[-10, -1];y=[-2, 10];z=[4, 0];same norm=l1

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

x to z difference coordinate0 := sub(-10, 4)

x to z difference coordinate0 = -14

x to z difference coordinate1 := sub(-1, 0)

x to z difference coordinate1 = -1

x to z:norm=l1;left=[-10, -1];right=[4, 0]

signed left-right=[-14, -1];use same named norm

x to z:coordinate=0;signed value=-14

absolute value=14

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

absolute value=1

x to z l1 sum := sum(14, 1)

x to z l1 sum = 15

x to y difference coordinate0 := sub(-10, -2)

x to y difference coordinate0 = -8

x to y difference coordinate1 := sub(-1, 10)

x to y difference coordinate1 = -11

x to y:norm=l1;left=[-10, -1];right=[-2, 10]

signed left-right=[-8, -11];use same named norm

x to y:coordinate=0;signed value=-8

absolute value=8

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

absolute value=11

x to y l1 sum := sum(8, 11)

x to y l1 sum = 19

y to z difference coordinate0 := sub(-2, 4)

y to z difference coordinate0 = -6

y to z difference coordinate1 := sub(10, 0)

y to z difference coordinate1 = 10

y to z:norm=l1;left=[-2, 10];right=[4, 0]

signed left-right=[-6, 10];use same named norm

y to z:coordinate=0;signed value=-6

absolute value=6

y to z:coordinate=1;signed value=10

absolute value=10

y to z l1 sum := sum(6, 10)

y to z l1 sum = 16

triangle right hand sum := sum(19, 16)

triangle right hand sum = 35

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, 15, 19, 16, 35, 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.