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An answer to the question 2 (written with K Sheshadri)

The guess made above that needs to be proved: For a general convex polygonal region with no symmetries, for every orientation, we have a unique line with that orientation that separates out the same fraction of both area and outer boundary length. The value of this common fraction of area and perimeter separated out will vary continuously with orientation.

Proof : Consider the convex polygonal region $C$ and a given orientation (direction). Draw both tangents to $C$ in that orientation. We assume both these tangents to touch $C$ at a single vertex (coincidences of tangents with entire edges of $C$ can be dealt with by small perturbations). Let these parallel tangents be distance $D$ apart. By sliding a line coincident with one of the tangents perpendicular to itself until it coincides with the other tangent to $C$, we get a continuous range of cutting lines. Let these cutting lines be parametrized by $d$, the perpendicular distance from the tangent from which we began sliding the cutting line.

For each value of $d$, we have a line that cuts $C$. Plot against $d$, the fraction of area Af of the full $C$ that the piece separated from $C$ has and also the fraction of perimeter Pf for the same piece. Obviously, as $d$ goes from 0 to $D$ both Af and Pf go from 0 to 1.

Now, we observe that the plot of Af has a quadratic behavior at both ends. Its plot will be continuous and made of several parabolic segments - beginning with an upward parabolic piece (where, as $d$ starts from 0, Af also starts from 0) and ending with a downward parabolic piece (when Af tends to 1 as $d$ approaches $D$). Moreover, due to convexity of $C$, the curve of Af rises monotonically.

On the other hand, Pf has a linear growth throughout including at ends. The graph is a continuous polyline and also rises monotonically.

From the above observations, as $d$ is increased from 0, the Af curve (quadratic) begins lower than Pf (linear) curve and as $d$ tends to $D$, Af approaches 1 from above the Pf curve. This plus the monotonically rising nature of both graphs plus their start values both being 0 and end values both being 1 guarantee that they have to necessarily intersect at some intermediate value of $d$; at these intersections, obviously, Af = Pf. It appears that convexity of $C$ also guarantees there will be only one such intersection.

Thus we have, for every orientation, a value of $d$ for which Af and Pf have same value - as claimed. If $C$ is centrally symmetric (circle, ellipse, rectangle, regular polygon with even number of sides...), the only such value of $d$ is $D$/2 and the common value of the fractions is 1/2 for all orientations. This will not be the case for asymmetric convex polygonal $C$'s - we have different common Af and Pf values for different orientations. This fraction should change continuously with orientation.

We guess that the envelope etc. of the cutting lines with common Af and Pf for each orientation might have interesting properties.

An answer to the question 2 (written with K Sheshadri)

The guess made above that needs to be proved: For a general convex polygonal region with no symmetries, for every orientation, we have a unique line with that orientation that separates out the same fraction of both area and outer boundary length. The value of this common fraction of area and perimeter separated out will vary continuously with orientation.

Proof : Consider the convex polygonal region $C$ and a given orientation (direction). Draw both tangents to $C$ in that orientation. We assume both these tangents to touch $C$ at a single vertex (coincidences of tangents with entire edges of $C$ can be dealt with by small perturbations). Let these parallel tangents be distance $D$ apart. By sliding a line coincident with one of the tangents perpendicular to itself until it coincides with the other tangent to $C$, we get a continuous range of cutting lines. Let these cutting lines be parametrized by $d$, the perpendicular distance from the tangent from which we began sliding the cutting line.

For each value of $d$, we have a line that cuts $C$. Plot against $d$, the fraction of area (call this fraction Af) of the full $C$ that the piece separated from $C$ has and also the fraction of perimeter (call this fraction Pf) for the same piece. Obviously, as $d$ goes from 0 to $D$ both Af and Pf go from 0 to 1.

Now, we observe that the plot of Af against d has a quadratic behavior at both ends. Its plot will be continuous and made of several parabolic segments - beginning with an upward parabolic piece (where, as $d$ starts from 0, Af also starts from 0) and ending with a downward parabolic piece (when Af tends to 1 as $d$ approaches $D$). Moreover, due to convexity of $C$, the curve of Af rises monotonically.

On the other hand, Pf has a linear behaviour throughout including at ends. This graph is a continuous polyline and also rises monotonically.

From the above observations, as $d$ is increased from 0, the Af curve (quadratic) begins lower than Pf (linear) curve and as $d$ tends to $D$, Af approaches 1 from above the Pf curve. This plus the monotonically rising nature of both graphs plus their start values both being 0 and end values both being 1 guarantee that they have to necessarily intersect at some intermediate value of $d$; at these intersections, obviously, Af = Pf. It appears that convexity of $C$ also guarantees there will be only one such intersection.

Thus we have, for every orientation, a value of $d$ for which Af and Pf have same value - as claimed. If $C$ is centrally symmetric (circle, ellipse, rectangle, regular polygon with even number of sides...), the only such value of $d$ is $D$/2 and the common value of the fractions is 1/2 for all orientations. This will not be the case for asymmetric convex polygonal $C$'s - we have different common Af and Pf values for different orientations. This fraction should change continuously with orientation.

We guess that the envelope etc. of the cutting lines with common Af and Pf for each orientation might have interesting properties.

An answer to the question 2 (written with K Sheshadri)

The guess made above that needs to be proved: For a general convex polygonal region with no symmetries, for every orientation, we have a unique line with that orientation that separates out the same fraction of both area and outer boundary length. The value of this common fraction of area and perimeter separated out will vary continuously with orientation.

Proof : Consider the convex polygonal region C and a given orientation (direction). Draw both tangents to C in that orientation. We assume both these tangents to touch C at a single vertex (coincidences of tangents with entire edges of C can be dealt with by small perturbations). Let these parallel tangents be distance D apart. By sliding a line coincident with one of the tangents perpendicular to itself until it coincides with the other tangent to C, we get a continuous range of cutting lines. Let them be parametrized by t, the perpendicular distance from the tangent from which we began sliding the cutting line.

For each value of t, we have a line that cuts C. Plot against t, the fraction of area Af of the full C that the piece separated from C has and also the fraction of perimeter Pf for the same piece. Obviously, as t goes from 0 to D, both Af and Pf go from 0 to 1.

Now, we observe that the plot of Af has a quadratic behavior at both ends. Its plot will be continuous and made of several parabolic segments - with an upward parabolic piece for t starting from 0 and a downward parabolic segment for t approaching 1. Moreover, due to convexity of C, the curve of Af rises monotonically.

On the other hand, Pf has a linear growth throughout including at ends. The graph is a continuous polyline and also rises monotonically.

From the above observations, as t is increased from 0, the Af curve (quadratic) begins lower than Pf (linear) curve and as t reaches D, Af approaches 1 from above the Pf curve. This plus the monotonically rising nature of both plus their start and end values both being 1 guarantee that they have to necessarily intersect at some intermediate value of t; at these intersections, obviously, Af = Pf. It appears that convexity of C guarantees there will be only one such intersection.

Thus we have, for every orientation, a value of t for which Af and Pf have same value as desired. If C is centrally symmetric (circle, ellipse, rectangle, regular polygon with even number of sides...), the only such value of t is D/2 and the common value of the fractions is 1/2 for all orientations. This will not be the case for asymmetric polygonal C's - we have different common Af and Pf values for different orientations. This fraction should change continuously with orientation.

We guess that the envelope etc of the cutting lines with common Af and Pf for each orientation might have interesting properties.

An answer to the question 2 (written with K Sheshadri)

The guess made above that needs to be proved: For a general convex polygonal region with no symmetries, for every orientation, we have a unique line with that orientation that separates out the same fraction of both area and outer boundary length. The value of this common fraction of area and perimeter separated out will vary continuously with orientation.

Proof : Consider the convex polygonal region $C$ and a given orientation (direction). Draw both tangents to $C$ in that orientation. We assume both these tangents to touch $C$ at a single vertex (coincidences of tangents with entire edges of $C$ can be dealt with by small perturbations). Let these parallel tangents be distance $D$ apart. By sliding a line coincident with one of the tangents perpendicular to itself until it coincides with the other tangent to $C$, we get a continuous range of cutting lines. Let these cutting lines be parametrized by $d$, the perpendicular distance from the tangent from which we began sliding the cutting line.

For each value of $d$, we have a line that cuts $C$. Plot against $d$, the fraction of area Af of the full $C$ that the piece separated from $C$ has and also the fraction of perimeter Pf for the same piece. Obviously, as $d$ goes from 0 to $D$ both Af and Pf go from 0 to 1.

Now, we observe that the plot of Af has a quadratic behavior at both ends. Its plot will be continuous and made of several parabolic segments - beginning with an upward parabolic piece (where, as $d$ starts from 0, Af also starts from 0) and ending with a downward parabolic piece (when Af tends to 1 as $d$ approaches $D$). Moreover, due to convexity of $C$, the curve of Af rises monotonically.

On the other hand, Pf has a linear growth throughout including at ends. The graph is a continuous polyline and also rises monotonically.

From the above observations, as $d$ is increased from 0, the Af curve (quadratic) begins lower than Pf (linear) curve and as $d$ tends to $D$, Af approaches 1 from above the Pf curve. This plus the monotonically rising nature of both graphs plus their start values both being 0 and end values both being 1 guarantee that they have to necessarily intersect at some intermediate value of $d$; at these intersections, obviously, Af = Pf. It appears that convexity of $C$ also guarantees there will be only one such intersection.

Thus we have, for every orientation, a value of $d$ for which Af and Pf have same value - as claimed. If $C$ is centrally symmetric (circle, ellipse, rectangle, regular polygon with even number of sides...), the only such value of $d$ is $D$/2 and the common value of the fractions is 1/2 for all orientations. This will not be the case for asymmetric convex polygonal $C$'s - we have different common Af and Pf values for different orientations. This fraction should change continuously with orientation.

We guess that the envelope etc. of the cutting lines with common Af and Pf for each orientation might have interesting properties.

A qualitative answer to the question 2 (written with K Sheshadri)

The guess made above that needs to be proved: For a general convex polygonal region with no symmetries, for every orientation, we have a unique line with that orientation that separates out the same fraction of both area and outer boundary length. The value of this common fraction of area and perimeter separated out will vary continuously with orientation.

Proof : Consider the convex polygonal region C and a given orientation (direction). Draw both tangents to C in that orientation. We assume both these tangents to touch C at a single vertex (coincidences of tangents with entire edges of C can be dealt with by small perturbations). Let these parallel tangents be distance D apart. By sliding a line coincident with one of the tangents perpendicular to itself until it coincides with the other tangent to C, we get a continuous range of cutting lines. Let them be parametrized by t, the perpendicular distance from the tangent from which we began sliding the cutting line.

For each value of t, we have a line that cuts C. Plot against t, the fraction of area Af of the full C that the piece separated from C has and also the fraction of perimeter Pf for the same piece. Obviously, as t goes from 0 to D, both Af and Pf go from 0 to 1.

Now, we observe that the plot of Af has a quadratic behavior at both ends. Its plot will be continuous and made of several parabolic segments - with an upward parabolic piece for t starting from 0 and a downward parabolic segment for t approaching 1. Moreover, due to convexity of C, the curve of Af rises monotonically.

On the other hand, Pf has a linear growth throughout including at ends. The graph is a continuous polyline and also rises monotonically.

From the above observations, as t is increased from 0, the Af curve (quadratic) begins lower than Pf (linear) curve and as t reaches D, Af approaches 1 from above the Pf curve. This plus the monotonically rising nature of both plus their start and end values both being 1 guarantee that they have to necessarily intersect at some intermediate value of t; at these intersections, obviously, Af = Pf. It appears that convexity of C guarantees there will be only one such intersection.

Thus we have, for every orientation, a value of t for which Af and Pf have same value as desired. If C is centrally symmetric (circle, ellipse, rectangle, regular polygon with even number of sides...), the only such value of t is D/2 and the common value of the fractions is 1/2 for all orientations. This will not be the case for asymmetric polygonal C's - we have different common Af and Pf values for different orientations. This fraction should change continuously with orientation.

We guess that the envelope etc of the cutting lines with common Af and Pf for each orientation might have interesting properties.

An answer to the question 2 (written with K Sheshadri)

The guess made above that needs to be proved: For a general convex polygonal region with no symmetries, for every orientation, we have a unique line with that orientation that separates out the same fraction of both area and outer boundary length. The value of this common fraction of area and perimeter separated out will vary continuously with orientation.

Proof : Consider the convex polygonal region C and a given orientation (direction). Draw both tangents to C in that orientation. We assume both these tangents to touch C at a single vertex (coincidences of tangents with entire edges of C can be dealt with by small perturbations). Let these parallel tangents be distance D apart. By sliding a line coincident with one of the tangents perpendicular to itself until it coincides with the other tangent to C, we get a continuous range of cutting lines. Let them be parametrized by t, the perpendicular distance from the tangent from which we began sliding the cutting line.

For each value of t, we have a line that cuts C. Plot against t, the fraction of area Af of the full C that the piece separated from C has and also the fraction of perimeter Pf for the same piece. Obviously, as t goes from 0 to D, both Af and Pf go from 0 to 1.

Now, we observe that the plot of Af has a quadratic behavior at both ends. Its plot will be continuous and made of several parabolic segments - with an upward parabolic piece for t starting from 0 and a downward parabolic segment for t approaching 1. Moreover, due to convexity of C, the curve of Af rises monotonically.

On the other hand, Pf has a linear growth throughout including at ends. The graph is a continuous polyline and also rises monotonically.

From the above observations, as t is increased from 0, the Af curve (quadratic) begins lower than Pf (linear) curve and as t reaches D, Af approaches 1 from above the Pf curve. This plus the monotonically rising nature of both plus their start and end values both being 1 guarantee that they have to necessarily intersect at some intermediate value of t; at these intersections, obviously, Af = Pf. It appears that convexity of C guarantees there will be only one such intersection.

Thus we have, for every orientation, a value of t for which Af and Pf have same value as desired. If C is centrally symmetric (circle, ellipse, rectangle, regular polygon with even number of sides...), the only such value of t is D/2 and the common value of the fractions is 1/2 for all orientations. This will not be the case for asymmetric polygonal C's - we have different common Af and Pf values for different orientations. This fraction should change continuously with orientation.

We guess that the envelope etc of the cutting lines with common Af and Pf for each orientation might have interesting properties.