{
  "first_figure": {
    "kernel": "delta_(0,0)+(1/5) delta_0 tensor g; g positive on(2,3)",
    "hull_segment": [
      [
        0,
        0
      ],
      [
        0,
        3
      ]
    ],
    "known_point_carrier": [
      0,
      0
    ],
    "hull_itself_not_claimed_to_be_an_actual_carrier": true,
    "X1_open_rectangle": [
      [
        -2,
        2
      ],
      [
        -4,
        1
      ]
    ],
    "X2_open_rectangle": [
      [
        -2,
        2
      ],
      [
        -1,
        1
      ]
    ],
    "point_carrier_translation_domain": "X1",
    "hull_translation_domain": "X2",
    "compact_image_bound": [
      [
        0,
        -1
      ],
      [
        0,
        0
      ]
    ],
    "outside_equation_point": [
      0,
      -2
    ],
    "sampled_witnesses": [
      {
        "j": 1,
        "point": [
          0,
          -0.5
        ]
      },
      {
        "j": 2,
        "point": [
          0,
          -0.75
        ]
      },
      {
        "j": 3,
        "point": [
          0,
          -0.875
        ]
      },
      {
        "j": 4,
        "point": [
          0,
          -0.9375
        ]
      },
      {
        "j": 5,
        "point": [
          0,
          -0.96875
        ]
      }
    ],
    "excluded_equation_boundary": [
      0,
      -1
    ],
    "lower_outward_normal": [
      0,
      -1
    ],
    "reflected_normal": [
      0,
      1
    ],
    "reflected_support_values": {
      "point_carrier": 0,
      "hull": 3
    },
    "proof_locators": [
      "formal Theorem4.1",
      "formal Theorem6.1",
      "learner Example2",
      "learner Exercise5"
    ]
  },
  "second_figure": {
    "boundary_formula": "s=abs(t)+t^2",
    "open_domain_formula": "s>abs(t)+t^2",
    "regular_tangent_parameters": [
      -1.0,
      -0.5,
      0.5,
      1.0
    ],
    "tangent_formula": "s=(sign(a)+2a)t-a^2; a!=0",
    "corner": [
      0,
      0
    ],
    "corner_excluded": true,
    "corner_tangent_gap_formula": "a^2>0",
    "normal_formula": "(sign(a)+2a,-1)/sqrt(1+(sign(a)+2a)^2)",
    "normal_arrow_parameters": [
      -0.5,
      0.5
    ],
    "normal_arrow_length": 0.45,
    "normal_plane_axes_have_equal_Euclidean_scale": true,
    "all_regular_tangents_strict_at_corner": true,
    "closed_halfspace_intersection": "s>=abs(t)+t^2",
    "interior_required_to_recover_open_domain": true,
    "proof_locators": [
      "formal Lemma3.1",
      "formal Lemma3.2",
      "formal Theorem7.1",
      "learner Example4",
      "learner Exercise7"
    ]
  }
}